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Quantitative microbiological risk assessment [ 1 ] (QMRA) is the process of estimating the risk from exposure to microorganisms .
The process involves measuring known microbial pathogens or indicators and running a Monte Carlo simulation to estimate the risk of transfer. [ 1 ] If a dose-response model is available for the microbe , it be used to estimate the probability of infection.
QMRA has expanded to be used to estimate microbial risk in many fields, but is particularly important in assessments of food [ 2 ] water supply [ 3 ] and human faeces/wastewater safety. [ 4 ]
The World Health Organisation's 2006 Guidelines for the Safe Use of Wastewater, Excreta and Greywater in Agriculture suggest that QMRA should be used to determine possible risk levels which can be achieved by sanitation systems. [ 5 ]
This microbiology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantitative_microbiological_risk_assessment |
Quantitative phase contrast microscopy or quantitative phase imaging are the collective names for a group of microscopy methods that quantify the phase shift that occurs when light waves pass through a more optically dense object. [ 1 ] [ 2 ] [ 3 ]
Translucent objects, like a living human cell, absorb and scatter small amounts of light.
This makes translucent objects much easier to observe in ordinary light microscopes.
Such objects do, however, induce a phase shift that can be observed using a phase contrast microscope .
Conventional phase contrast microscopy and related methods, such as differential interference contrast microscopy , visualize phase shifts by transforming phase shift gradients into intensity variations.
These intensity variations are mixed with other intensity variations, making it difficult to extract quantitative information.
Quantitative phase contrast methods are distinguished from conventional phase contrast methods in that
they create a second so-called phase shift image or phase image , independent of the intensity ( bright field ) image. Phase unwrapping methods are generally applied to the phase shift image to give absolute phase shift
values in each pixel, as exemplified by Figure 1.
The principal methods for measuring and visualizing phase shifts include ptychography and various types
of holographic microscopy methods such as digital holographic microscopy , holographic interference microscopy and digital in-line holographic microscopy.
Common to these methods is that an interference pattern ( hologram ) is recorded by a digital image sensor .
From the recorded interference pattern, the intensity and the phase shift image is numerically created by a computer algorithm . [ 5 ]
Quantitative phase contrast microscopy is primarily used to observe unstained living cells.
Measuring the phase delay images of biological cells provides quantitative information about the morphology and drymass of individual cells. [ 6 ] These features can be analyzed with image analysis software, which has led to the development of non-invasive live cell imaging and automated cell culture analysis systems based on quantitative phase contrast microscopy. [ 7 ] | https://en.wikipedia.org/wiki/Quantitative_phase-contrast_microscopy |
Quantitative precipitation estimation or QPE is a method of approximating the amount of precipitation that has fallen at a location or across a region. Maps of the estimated amount of precipitation to have fallen over a certain area and time span are compiled using several different data sources including manual and automatic field observations and radar and satellite data. This process is undertaken every day across the United States at Weather Forecast Offices (WFOs) run by the National Weather Service (NWS).
A number of different algorithms can be used to estimate precipitation amounts from data collected by radar, satellites, or other remote sensing platforms. [ 1 ] Research in the fields of QPE and quantitative precipitation forecasting (QPF) is ongoing.
Recent research in the field suggests using commercial microwave links for environmental monitoring in general and precipitation measurements in particular. [ 2 ]
This meteorology –related article is a stub . You can help Wikipedia by expanding it .
This hydrology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantitative_precipitation_estimation |
Quantitative storytelling ( QST ) is a systematic approach to exploring the many frames potentially legitimate in a scientific study or controversy . [ 1 ] QST assumes that, in an interconnected society, multiple frameworks and worldviews are legitimately upheld by different entities and social actors. QST looks critically at models used in evidence-based policy . Such models are often reductionist in that tractability (i.e. the possibility of proceeding towards a solution to a given problem) is achieved at the expense of suppressing available evidence. [ 2 ] QST suggests corrective approaches to this practice.
Quantitative storytelling (QST) addresses evidence-based policy and can be considered a reaction to methods of quantification with cost-benefit analysis or risk analysis .
Jerome Ravetz [ 3 ] and Steve Rayner [ 4 ] discuss the concept that some of the evidence needed for policy is removed from view. They suggest that 'uncomfortable knowledge' is subtracted from the policy discourse with the objective of easing tractability or to advance a given agenda. The word ' hypo-cognition ' has been used in the context of these instrumental uses of frames. [ 5 ] [ 6 ]
According to Rayner , a phenomenon of "displacement" takes place when a model becomes the objective instead of the tool, for example, when an institution chooses to monitor and manage the outcome of a model rather than what happens in reality. [ 4 ] Once exposed, the strategic use of hypocognition erodes the trust in the involved actors and institutions. [ 4 ]
QST suggests acknowledging ignorance, as to work out 'clumsy solutions', [ 4 ] which may permit negotiation to be had among parties with different normative orientations. QST is also sensitive to power and knowledge asymmetries, [ 7 ] [ 8 ] as interest groups have more scope to capture regulators than the average citizen and consumer. [ citation needed ]
QST does not forbid the use of quantitative tools altogether. It suggests instead to quantitatively explore multiple narratives, avoiding spurious accuracy and focusing on some salient features of the selected stories. Rather than attempting to amass evidence in support of a given reading or policy, or to optimise it with modelling, QST tries to test whether a given policy option or framing conflicts with existing social or biophysical constraints. These are: [ 9 ]
A recent application of QST exploring the transition to intermittent electrical energy supply in has been made in Germany and Spain . [ 10 ] They use QST to explore a case of water and agricultural governance in the Canary Islands . [ 11 ] | https://en.wikipedia.org/wiki/Quantitative_storytelling |
Quantitative structure–activity relationship models ( QSAR models) are regression or classification models used in the chemical and biological sciences and engineering. Like other regression models, QSAR regression models relate a set of "predictor" variables (X) to the potency of the response variable (Y), while classification QSAR models relate the predictor variables to a categorical value of the response variable.
In QSAR modeling, the predictors consist of physico-chemical properties or theoretical molecular descriptors [ 1 ] [ 2 ] of chemicals; the QSAR response-variable could be a biological activity of the chemicals. QSAR models first summarize a supposed relationship between chemical structures and biological activity in a data-set of chemicals. Second, QSAR models predict the activities of new chemicals. [ 3 ] [ 4 ]
Related terms include quantitative structure–property relationships ( QSPR ) when a chemical property is modeled as the response variable. [ 5 ] [ 6 ] "Different properties or behaviors of chemical molecules have been investigated in the field of QSPR. Some examples are quantitative structure–reactivity relationships (QSRRs), quantitative structure–chromatography relationships (QSCRs) and, quantitative structure–toxicity relationships (QSTRs), quantitative structure–electrochemistry relationships (QSERs), and quantitative structure– biodegradability relationships (QSBRs)." [ 7 ]
As an example, biological activity can be expressed quantitatively as the concentration of a substance required to give a certain biological response. Additionally, when physicochemical properties or structures are expressed by numbers, one can find a mathematical relationship, or quantitative structure-activity relationship, between the two. The mathematical expression, if carefully validated, [ 8 ] [ 9 ] [ 10 ] [ 11 ] can then be used to predict the modeled response of other chemical structures. [ 12 ]
A QSAR has the form of a mathematical model :
The error includes model error ( bias ) and observational variability, that is, the variability in observations even on a correct model.
The principal steps of QSAR/QSPR include: [ 7 ]
The basic assumption for all molecule-based hypotheses is that similar molecules have similar activities. This principle is also called Structure–Activity Relationship ( SAR ). The underlying problem is therefore how to define a small difference on a molecular level, since each kind of activity, e.g. reaction ability, biotransformation ability, solubility , target activity, and so on, might depend on another difference. Examples were given in the bioisosterism reviews by Patanie/LaVoie [ 13 ] and Brown. [ 14 ]
In general, one is more interested in finding strong trends . Created hypotheses usually rely on a finite number of chemicals, so care must be taken to avoid overfitting : the generation of hypotheses that fit training data very closely but perform poorly when applied to new data.
The SAR paradox refers to the fact that it is not the case that all similar molecules have similar activities [ citation needed ] .
Analogously, the " partition coefficient "—a measurement of differential solubility and itself a component of QSAR predictions—can be predicted either by atomic methods (known as "XLogP" or "ALogP") or by chemical fragment methods (known as "CLogP" and other variations). It has been shown that the logP of compound can be determined by the sum of its fragments; fragment-based methods are generally accepted as better predictors than atomic-based methods. [ 15 ] Fragmentary values have been determined statistically, based on empirical data for known logP values. This method gives mixed results and is generally not trusted to have accuracy of more than ±0.1 units. [ 16 ]
Group or fragment-based QSAR is also known as GQSAR. [ 17 ] GQSAR allows flexibility to study various molecular fragments of interest in relation to the variation in biological response. The molecular fragments could be substituents at various substitution sites in congeneric set of molecules or could be on the basis of pre-defined chemical rules in case of non-congeneric sets. GQSAR also considers cross-terms fragment descriptors, which could be helpful in identification of key fragment interactions in determining variation of activity. [ 17 ] Lead discovery using fragnomics is an emerging paradigm. In this context FB-QSAR proves to be a promising strategy for fragment library design and in fragment-to-lead identification endeavours. [ 18 ]
An advanced approach on fragment or group-based QSAR based on the concept of pharmacophore-similarity is developed. [ 19 ] This method, pharmacophore-similarity-based QSAR (PS-QSAR) uses topological pharmacophoric descriptors to develop QSAR models. This activity prediction may assist the contribution of certain pharmacophore features encoded by respective fragments toward activity improvement and/or detrimental effects. [ 19 ]
The acronym 3D-QSAR or 3-D QSAR refers to the application of force field calculations requiring three-dimensional structures of a given set of small molecules with known activities (training set). The training set needs to be superimposed (aligned) by either experimental data (e.g. based on ligand-protein crystallography ) or molecule superimposition software. It uses computed potentials, e.g. the Lennard-Jones potential , rather than experimental constants and is concerned with the overall molecule rather than a single substituent. The first 3-D QSAR was named Comparative Molecular Field Analysis (CoMFA) by Cramer et al. It examined the steric fields (shape of the molecule) and the electrostatic fields [ 20 ] which were correlated by means of partial least squares regression (PLS).
The created data space is then usually reduced by a following feature extraction (see also dimensionality reduction ). The following learning method can be any of the already mentioned machine learning methods, e.g. support vector machines . [ 21 ] An alternative approach uses multiple-instance learning by encoding molecules as sets of data instances, each of which represents a possible molecular conformation. A label or response is assigned to each set corresponding to the activity of the molecule, which is assumed to be determined by at least one instance in the set (i.e. some conformation of the molecule). [ 22 ]
On June 18, 2011 the Comparative Molecular Field Analysis (CoMFA) patent has dropped any restriction on the use of GRID and partial least-squares (PLS) technologies. [ citation needed ]
In this approach, descriptors quantifying various electronic, geometric, or steric properties of a molecule are computed and used to develop a QSAR. [ 23 ] This approach is different from the fragment (or group contribution) approach in that the descriptors are computed for the system as whole rather than from the properties of individual fragments. This approach is different from the 3D-QSAR approach in that the descriptors are computed from scalar quantities (e.g., energies, geometric parameters) rather than from 3D fields.
An example of this approach is the QSARs developed for olefin polymerization by half sandwich compounds . [ 24 ] [ 25 ]
It has been shown that activity prediction is even possible based purely on the SMILES string. [ 26 ] [ 27 ] [ 28 ]
Similarly to string-based methods, the molecular graph can directly be used as input for QSAR models, [ 29 ] [ 30 ] but usually yield inferior performance compared to descriptor-based QSAR models. [ 31 ] [ 32 ]
QSAR has been merged with the similarity-based read-across technique to develop a new field of q-RASAR . The DTC Laboratory at Jadavpur University has developed this hybrid method and the details are available at their laboratory page . Recently, the q-RASAR framework has been improved by its integration with the ARKA descriptors in QSAR .
In the literature it can be often found that chemists have a preference for partial least squares (PLS) methods, [ citation needed ] since it applies the feature extraction and induction in one step.
Computer SAR models typically calculate a relatively large number of features. Because those lack structural interpretation ability, the preprocessing steps face a feature selection problem (i.e., which structural features should be interpreted to determine the structure-activity relationship). Feature selection can be accomplished by visual inspection (qualitative selection by a human); by data mining; or by molecule mining.
A typical data mining based prediction uses e.g. support vector machines , decision trees , artificial neural networks for inducing a predictive learning model.
Molecule mining approaches, a special case of structured data mining approaches, apply a similarity matrix based prediction or an automatic fragmentation scheme into molecular substructures. Furthermore, there exist also approaches using maximum common subgraph searches or graph kernels . [ 33 ] [ 34 ]
Typically QSAR models derived from non linear machine learning is seen as a "black box", which fails to guide medicinal chemists. Recently there is a relatively new concept of matched molecular pair analysis [ 35 ] or prediction driven MMPA which is coupled with QSAR model in order to identify activity cliffs. [ 36 ]
QSAR modeling produces predictive models derived from application of statistical tools correlating biological activity (including desirable therapeutic effect and undesirable side effects) or physico-chemical properties in QSPR models of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of molecular structure or properties . QSARs are being applied in many disciplines, for example: risk assessment , toxicity prediction, and regulatory decisions [ 37 ] in addition to drug discovery and lead optimization . [ 38 ] Obtaining a good quality QSAR model depends on many factors, such as the quality of input data, the choice of descriptors and statistical methods for modeling and for validation. Any QSAR modeling should ultimately lead to statistically robust and predictive models capable of making accurate and reliable predictions of the modeled response of new compounds.
For validation of QSAR models, usually various strategies are adopted: [ 39 ]
The success of any QSAR model depends on accuracy of the input data, selection of appropriate descriptors and statistical tools, and most importantly validation of the developed model. Validation is the process by which the reliability and relevance of a procedure are established for a specific purpose; for QSAR models validation must be mainly for robustness, prediction performances and applicability domain (AD) of the models. [ 8 ] [ 9 ] [ 11 ] [ 40 ] [ 41 ]
Some validation methodologies can be problematic. For example, leave one-out cross-validation generally leads to an overestimation of predictive capacity. Even with external validation, it is difficult to determine whether the selection of training and test sets was manipulated to maximize the predictive capacity of the model being published.
Different aspects of validation of QSAR models that need attention include methods of selection of training set compounds, [ 42 ] setting training set size [ 43 ] and impact of variable selection [ 44 ] for training set models for determining the quality of prediction. Development of novel validation parameters for judging quality of QSAR models is also important. [ 11 ] [ 45 ] [ 46 ]
One of the first historical QSAR applications was to predict boiling points . [ 47 ]
It is well known for instance that within a particular family of chemical compounds , especially of organic chemistry , that there are strong correlations between structure and observed properties. A simple example is the relationship between the number of carbons in alkanes and their boiling points . There is a clear trend in the increase of boiling point with an increase in the number carbons, and this serves as a means for predicting the boiling points of higher alkanes .
A still very interesting application is the Hammett equation , Taft equation and pKa prediction methods. [ 48 ]
The biological activity of molecules is usually measured in assays to establish the level of inhibition of particular signal transduction or metabolic pathways . Drug discovery often involves the use of QSAR to identify chemical structures that could have good inhibitory effects on specific targets and have low toxicity (non-specific activity). Of special interest is the prediction of partition coefficient log P , which is an important measure used in identifying " druglikeness " according to Lipinski's Rule of Five . [ citation needed ]
While many quantitative structure activity relationship analyses involve the interactions of a family of molecules with an enzyme or receptor binding site, QSAR can also be used to study the interactions between the structural domains of proteins. Protein-protein interactions can be quantitatively analyzed for structural variations resulted from site-directed mutagenesis . [ 49 ]
It is part of the machine learning method to reduce the risk for a SAR paradox, especially taking into account that only a finite amount of data is available (see also MVUE ). In general, all QSAR problems can be divided into coding [ 50 ] and learning . [ 51 ]
(Q)SAR models have been used for risk management . QSARS are suggested by regulatory authorities; in the European Union , QSARs are suggested by the REACH regulation, where "REACH" abbreviates "Registration, Evaluation, Authorisation and Restriction of Chemicals". Regulatory application of QSAR methods includes in silico toxicological assessment of genotoxic impurities. [ 52 ] Commonly used QSAR assessment software such as DEREK or CASE Ultra (MultiCASE) is used to genotoxicity of impurity according to ICH M7. [ 53 ]
The chemical descriptor space whose convex hull is generated by a particular training set of chemicals is called the training set's applicability domain . Prediction of properties of novel chemicals that are located outside the applicability domain uses extrapolation , and so is less reliable (on average) than prediction within the applicability domain. The assessment of the reliability of QSAR predictions remains a research topic. [ citation needed ]
The QSAR equations can be used to predict biological activities of newer molecules before their synthesis.
Examples of machine learning tools for QSAR modeling include: [ 54 ] | https://en.wikipedia.org/wiki/Quantitative_structure–activity_relationship |
Quantitative systems pharmacology (QSP) is a discipline within biomedical research that uses mathematical computer models to characterize biological systems, disease processes and drug pharmacology. [ 1 ] [ 2 ] QSP can be viewed as a sub-discipline of pharmacometrics that focuses on modeling the mechanisms of drug pharmacokinetics (PK), pharmacodynamics (PD), and disease processes using a systems pharmacology point of view. QSP models are typically defined by systems of ordinary differential equations (ODE) that depict the dynamical properties of the interaction between the drug and the biological system.
QSP can be used to generate biological/pharmacological hypotheses in silico to aid in the design of in vitro or in vivo non-clinical and clinical experiments. This can help to guide biomedical experiments so that they yield more meaningful data. QSP is increasingly being used for this purpose in pharmaceutical research & development to help guide the discovery and development of new therapies. [ 3 ] [ 4 ] QSP has been used by the FDA in a clinical pharmacology review. [ 5 ]
QSP emerged as a discipline through two workshops held at the National Institutes of Health (NIH) in 2008 and 2010, with the goal of merging of systems biology and pharmacology. The workshops outlined a need for a mathematical discipline to aid in translational medicine. QSP proposed integrating concepts, methods, and investigators from computational biology, systems biology, and biological engineering into pharmacology. [ 2 ]
A review of the history and future of QSP identified areas where it has advanced understanding of drug mechanisms, supported preclinical to clinical translation, and in general aided in drug development. The FDA has included QSP as a component of the Model-Informed Drug Development Program. [ 6 ] [ 7 ] | https://en.wikipedia.org/wiki/Quantitative_systems_pharmacology |
Quantities, Units and Symbols in Physical Chemistry , also known as the Green Book , is a compilation of terms and symbols widely used in the field of physical chemistry . It also includes a table of physical constants , tables listing the properties of elementary particles , chemical elements , and nuclides , and information about conversion factors that are commonly used in physical chemistry. The Green Book is published by the International Union of Pure and Applied Chemistry (IUPAC) and is based on published, citeable sources. Information in the Green Book is synthesized from recommendations made by IUPAC, the International Union of Pure and Applied Physics (IUPAP) and the International Organization for Standardization (ISO), including recommendations listed in the IUPAP Red Book Symbols, Units, Nomenclature and Fundamental Constants in Physics and in the ISO 31 standards.
The third edition of the Green Book ( ISBN 978-0-85404-433-7 ) was first published by IUPAC in 2007. A second printing of the third edition was released in 2008; this printing made several minor revisions to the 2007 text. A third printing of the third edition was released in 2011. The text of the third printing is identical to that of the second printing.
A Japanese translation of the third edition of the Green Book ( ISBN 978-4-06-154359-1 ) was published in 2009. A French translation of the third edition of the Green Book ( ISBN 978-2-8041-7207-7 ) was published in 2012. A Portuguese translation (Brazilian Portuguese and European Portuguese) of the third edition of the Green Book ( ISBN 978-85-64099-19-7 ) was published in 2018, with updated values of the physical constants and atomic weights; it is referred to as the "Livro Verde".
A concise four-page summary of the most important material in the Green Book was published in the July–August 2011 issue of Chemistry International , the IUPAC news magazine.
The second edition of the Green Book ( ISBN 0-632-03583-8 ) was first published in 1993. It was reprinted in 1995, 1996, and 1998.
The Green Book is a direct successor of the Manual of Symbols and Terminology for Physicochemical Quantities and Units , originally prepared for publication on behalf of IUPAC's Physical Chemistry Division by M. L. McGlashen in 1969. A full history of the Green Book's various editions is provided in the historical introduction to the third edition.
The second edition and the third edition (second printing) of the Green Book have both been made available online as PDF files; the PDF version of the third edition is fully searchable. The four-page concise summary is also available online as a PDF file. External Links (below) .
In addition to the obvious data on quantities, units and symbols, the compilation contains some less obvious but very useful information on related topics.
Unit conversion is a notorious source of errors. Many people apply individual rules, e.g. "to obtain length in centimeters multiply the length in inches by 2.54", but combining several such conversions is laborious and prone to mistakes. A better way is to use the factor-label method , which is closely related to dimensional analysis , and quantity calculus explained in sections 1.1 and 7.1 of this compilation.
Section 1.3 explains the rules for writing scientific symbols and names, for example, where to use capital letters or italics, and where their use is incorrect. The typographical rules are extensive, including even such detail as whether "20°C" or "20 °C" is the correct form.
Section 3.8 introduces atomic units and gives a table of atomic units of various physical quantities and the conversion factor to the SI units . Section 7.3(v) gives a concise but clear tutorial on practical use of atomic units, in particular how to understand equations "written in atomic units". | https://en.wikipedia.org/wiki/Quantities,_Units_and_Symbols_in_Physical_Chemistry |
Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities . [ 1 ] [ a ]
Its roots can be traced to Fourier's concept of dimensional analysis (1822). [ 2 ] The basic axiom of quantity calculus is Maxwell's description [ 3 ] of a physical quantity as the product of a "numerical value" and a "reference quantity" (i.e. a "unit quantity" or a " unit of measurement "). De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full axiomatization has yet to be completed. [ 1 ]
Measurements are expressed as products of a numeric value with a unit symbol, e.g. "12.7 m". Unlike algebra, the unit symbol represents a measurable quantity such as a metre, not an algebraic variable i.e. the unit symbol does not satisfy the axioms of arithmetic. [ 4 ]
A careful distinction needs to be made between abstract quantities and measurable quantities. The multiplication and division rules of quantity calculus are applied to SI base units (which are measurable quantities) to define SI derived units , including dimensionless derived units, such as the radian (rad) and steradian (sr) which are useful for clarity, although they are both algebraically equal to 1. Thus there is some disagreement about whether it is meaningful to multiply or divide units. Emerson suggests that if the units of a quantity are algebraically simplified, they then are no longer units of that quantity. [ 5 ] Johansson proposes that there are logical flaws in the application of quantity calculus, and that the so-called dimensionless quantities should be understood as "unitless quantities". [ 6 ]
How to use quantity calculus for unit conversion and keeping track of units in algebraic manipulations is explained in the handbook Quantities, Units and Symbols in Physical Chemistry . | https://en.wikipedia.org/wiki/Quantity_calculus |
In engineering and science , dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length , mass , time , and electric current ) and units of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.
Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.
Any physically meaningful equation , or inequality , must have the same dimensions on its left and right sides, a property known as dimensional homogeneity . Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations . It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.
The concept of physical dimension or quantity dimension , and of dimensional analysis, was introduced by Joseph Fourier in 1822. [ 1 ] : 42
The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the rank of the dimensional matrix . Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization , which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or physical constants of nature. [ 1 ] : 43 This may give insight into the fundamental properties of the system, as illustrated in the examples below.
The dimension of a physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally rational ) power . The dimension of a physical quantity is more fundamental than some scale or unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. Natural units , being based on only universal constants, may be thought of as being "less arbitrary".
There are many possible choices of base physical dimensions. The SI standard selects the following dimensions and corresponding dimension symbols :
The symbols are by convention usually written in roman sans serif typeface. [ 2 ] Mathematically, the dimension of the quantity Q is given by
where a , b , c , d , e , f , g are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge , since Q = TI .
A quantity that has only b ≠ 0 (with all other exponents zero) is known as a geometric quantity . A quantity that has only both a ≠ 0 and b ≠ 0 is known as a kinematic quantity . A quantity that has only all of a ≠ 0 , b ≠ 0 , and c ≠ 0 is known as a dynamic quantity . [ 3 ] A quantity that has all exponents null is said to have dimension one . [ 2 ]
The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm ; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.
There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, [ 4 ] although this does not invalidate the usefulness of dimensional analysis.
As examples, the dimension of the physical quantity speed v is
The dimension of the physical quantity acceleration a is
The dimension of the physical quantity force F is
The dimension of the physical quantity pressure P is
The dimension of the physical quantity energy E is
The dimension of the physical quantity power P is
The dimension of the physical quantity electric charge Q is
The dimension of the physical quantity voltage V is
The dimension of the physical quantity capacitance C is
In dimensional analysis, Rayleigh's method is a conceptual tool used in physics , chemistry , and engineering . It expresses a functional relationship of some variables in the form of an exponential equation . It was named after Lord Rayleigh .
The method involves the following steps:
As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.
Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number —a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with division , e.g. 60 km/h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition ), powers (like m 2 for square metres), or combinations thereof.
A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed. [ 5 ] For example, units for length and time are normally chosen as base units. Units for volume , however, can be factored into the base units of length (m 3 ), thus they are considered derived or compound units.
Sometimes the names of units obscure the fact that they are derived units. For example, a newton (N) is a unit of force , which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s −2 ). The newton is defined as 1 N = 1 kg⋅m⋅s −2 .
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100 .
Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
In economics, one distinguishes between stocks and flows : a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars/year).
In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
The most basic rule of dimensional analysis is that of dimensional homogeneity. [ 6 ]
However, the dimensions form an abelian group under multiplication, so:
For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.
The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if m man , m rat and L man denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression m man + m rat is meaningful, but the heterogeneous expression m man + L man is meaningless. However, m man / L 2 man is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions.
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension T −2 L 2 M , they are fundamentally different physical quantities.
To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m.
A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables. [ 7 ] For example, Newton's laws of motion must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor . For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar . The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1 . Since any quantity can be multiplied by 1 without changing it, the expression " 100 kPa / 1 bar " can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500 , and bar/bar cancels out, so 5 bar = 500 kPa .
Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n -ball (the solid ball in n dimensions), or the area of its surface, the n -sphere : being an n -dimensional figure, the volume scales as x n , while the surface area, being ( n − 1) -dimensional, scales as x n −1 . Thus the volume of the n -ball in terms of the radius is C n r n , for some constant C n . Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows . More generally, dimensional analysis is used in interpreting various financial ratios , economics ratios, and accounting ratios.
In fluid mechanics , dimensional analysis is performed to obtain dimensionless pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships. [ 8 ] In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
The origins of dimensional analysis have been disputed by historians. [ 9 ] [ 10 ] The first written application of dimensional analysis has been credited to François Daviet , a student of Joseph-Louis Lagrange , in a 1799 article at the Turin Academy of Science. [ 10 ]
This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the Buckingham π theorem . Simeon Poisson also treated the same problem of the parallelogram law by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39). [ 11 ] In the second edition of 1833, Poisson explicitly introduces the term dimension instead of the Daviet homogeneity .
In 1822, the important Napoleonic scientist Joseph Fourier made the first credited important contributions [ 12 ] based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables.
James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. [ 13 ] Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant G is taken as unity , thereby defining M = T −2 L 3 . [ 14 ] By assuming a form of Coulomb's law in which the Coulomb constant k e is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = T −1 L 3/2 M 1/2 , [ 15 ] which, after substituting his M = T −2 L 3 equation for mass, results in charge having the same dimensions as mass, viz. Q = T −2 L 3 .
Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by Lord Rayleigh , who was trying to understand why the sky is blue. [ 16 ] Rayleigh first published the technique in his 1877 book The Theory of Sound . [ 17 ]
The original meaning of the word dimension , in Fourier's Theorie de la Chaleur , was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. [ 18 ] This was slightly changed by Maxwell, who said the dimensions of acceleration are T −2 L, instead of just the exponents. [ 19 ]
What is the period of oscillation T of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g ? That period is the solution for T of some dimensionless equation in the variables T , m , k , and g .
The four quantities have the following dimensions: T [T]; m [M]; k [M/T 2 ]; and g [L/T 2 ]. From these we can form only one dimensionless product of powers of our chosen variables, G 1 = T 2 k / m [T 2 · M/T 2 / M = 1] , and putting G 1 = C for some dimensionless constant C gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical group . They are often called dimensionless numbers as well.
The variable g does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines g with k , m , and T , because g is the only quantity that involves the dimension L. This implies that in this problem the g is irrelevant. Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g : it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: T = κ m k {\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}} , for some dimensionless constant κ (equal to C {\displaystyle {\sqrt {C}}} from the original dimensionless equation).
When faced with a case where dimensional analysis rejects a variable ( g , here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.
When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as κ .
Consider the case of a vibrating wire of length ℓ (L) vibrating with an amplitude A (L). The wire has a linear density ρ (M/L) and is under tension s (LM/T 2 ), and we want to know the energy E (L 2 M/T 2 ) in the wire. Let π 1 and π 2 be two dimensionless products of powers of the variables chosen, given by
The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
where F is some unknown function, or, equivalently as
where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f . But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ℓ , and so infer that E = ℓs . The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.
The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number , which may be interpreted by dimensional analysis.
Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness t (L) and radius R (L). The disc has a density ρ (M/L 3 ), rotates at an angular velocity ω (T −1 ) and this leads to a stress S (T −2 L −1 M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following ( 5 − 3 = 2 ) non-dimensional groups:
Through the use of numerical experiments using, for example, the finite element method , the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs. [ 20 ]
The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an abelian group : The identity is written as 1; [ citation needed ] L 0 = 1 , and the inverse of L is 1/L or L −1 . L raised to any integer power p is a member of the group, having an inverse of L − p or 1/L p . The operation of the group is multiplication, having the usual rules for handling exponents ( L n × L m = L n + m ). Physically, 1/L can be interpreted as reciprocal length , and 1/T as reciprocal time (see reciprocal second ).
An abelian group is equivalent to a module over the integers, with the dimensional symbol T i L j M k corresponding to the tuple ( i , j , k ) . When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the module.
A basis for such a module of dimensional symbols is called a set of base quantities , and all other vectors are called derived units. As in any module, one may choose different bases , which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa).
The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module, (0, 0, 0) .
In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like V L 1/2 . [ 21 ] However, it is not possible to take arbitrary fractional powers of units, due to representation-theoretic obstructions. [ 22 ]
One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensions M and L , one has the vector spaces V M and V L , and can define V ML := V M ⊗ V L as the tensor product . Similarly, the dual space can be interpreted as having "negative" dimensions. [ 23 ] This corresponds to the fact that under the natural pairing between a vector space and its dual, the dimensions cancel, leaving a dimensionless scalar.
The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity describes some number (e.g., m ) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, {π 1 , ..., π m } . (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating ) together the measured quantities to produce something with the same unit as some derived quantity X can be expressed in the general form
Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form
Knowing this restriction can be a powerful tool for obtaining new insight into the system.
The dimension of physical quantities of interest in mechanics can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis . The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a basis : they must span the space, and be linearly independent .
For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T 2 ], L, M, while the latter can be expressed as [T = (LM/F) 1/2 ], L, M.
On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of electric charge . In thermodynamics , the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant , ≈ 6.02 × 10 23 mol −1 ) is also defined as a base dimension, N.
In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter , connected with the symmetry properties of the collisionless Vlasov equation , is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.
Bridgman's theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor. [ 24 ] [ 25 ] This excludes polynomials of more than one term or transcendental functions not of that form.
Scalar arguments to transcendental functions such as exponential , trigonometric and logarithmic functions, or to inhomogeneous polynomials , must be dimensionless quantities . (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)
While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log( a / b ) = log a − log b , where the logarithm is taken in any base, holds for dimensionless numbers a and b , but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not. [ 26 ]
Similarly, while one can evaluate monomials ( x n ) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x 2 , the expression (3 m) 2 = 9 m 2 makes sense (as an area), while for x 2 + x , the expression (3 m) 2 + 3 m = 9 m 2 + 3 m does not make sense.
However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,
This is the height to which an object rises in time t if the acceleration of gravity is 9.8 metres per second per second and the initial upward speed is 500 metres per second . It is not necessary for t to be in seconds . For example, suppose t = 0.01 minutes. Then the first term would be
The value of a dimensional physical quantity Z is written as the product of a unit [ Z ] within the dimension and a dimensionless numerical value or numerical factor, n . [ 27 ]
When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A conversion factor , which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
The factor 0.3048 m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.
Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.
A quantity equation , also sometimes called a complete equation , is an equation that remains valid independently of the unit of measurement used when expressing the physical quantities . [ 28 ]
In contrast, in a numerical-value equation , just the numerical values of the quantities occur, without units. Therefore, it is only valid when each numerical values is referenced to a specific unit.
For example, a quantity equation for displacement d as speed s multiplied by time difference t would be:
for s = 5 m/s, where t and d may be expressed in any units, converted if necessary.
In contrast, a corresponding numerical-value equation would be:
where T is the numeric value of t when expressed in seconds and D is the numeric value of d when expressed in metres.
Generally, the use of numerical-value equations is discouraged. [ 28 ]
The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the κ in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make " back of the envelope " calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, χ ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be ~ 1/ χ d , where d is the dimension of the lattice.
It has been argued by some physicists, e.g., Michael J. Duff , [ 4 ] [ 29 ] that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: c , ħ , and G , in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.
Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ , c , and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit c → ∞ , ħ → 0 and G → 0 . In problems involving a gravitational field the latter limit should be taken such that the field stays finite.
Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force. [ 30 ] [ 31 ] [ 32 ]
T −2 L 2 M
T −1 LM
T −2 LM
Dimensional correctness as part of type checking has been studied since 1977. [ 33 ] Implementations for Ada [ 34 ] and C++ [ 35 ] were described in 1985 and 1988.
Kennedy's 1996 thesis describes an implementation in Standard ML , [ 36 ] and later in F# . [ 37 ] There are implementations for Haskell , [ 38 ] OCaml , [ 39 ] and Rust , [ 40 ] Python, [ 41 ] and a code checker for Fortran . [ 42 ] [ 43 ] Griffioen's 2019 thesis extended Kennedy's Hindley–Milner type system to support Hart's matrices. [ 44 ] [ 45 ] McBride and Nordvall-Forsberg show how to use dependent types to extend type systems for units of measure. [ 46 ]
Mathematica 13.2 has a function for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation. [ 47 ] Mathematica also has a function to find the dimensions of a unit such as 1 J named UnitDimensions. [ 48 ] Mathematica also has a function that will find dimensionally equivalent combinations of a subset of physical quantities named DimensionalCombations. [ 49 ] Mathematica can also factor out certain dimension with UnitDimensions by specifying an argument to the function UnityDimensions. [ 50 ] For example, you can use UnityDimensions to factor out angles. [ 50 ] In addition to UnitDimensions, Mathematica can find the dimensions of a QuantityVariable with the function QuantityVariableDimensions. [ 51 ]
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors; [ citation needed ] vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin . While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).
Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:
This illustrates the subtle distinction between affine quantities (ones modeled by an affine space , such as position) and vector quantities (ones modeled by a vector space , such as displacement).
Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference , while to assign a number to a vector unit only requires a unit of measurement.
Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.
This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero,
where the symbol ≘ means corresponds to , since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.
For temperature differences,
(Here °R refers to the Rankine scale , not the Réaumur scale ).
Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.
Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction . (In 1 dimension, this issue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in multi-dimensional Euclidean space, one also needs a bearing: they need to be compared to a frame of reference .
This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.
Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank m {\displaystyle m} of the dimensional matrix. [ 52 ]
He introduced two approaches:
As an example of the usefulness of the first approach, suppose we wish to calculate the distance a cannonball travels when fired with a vertical velocity component v y {\displaystyle v_{\text{y}}} and a horizontal velocity component v x {\displaystyle v_{\text{x}}} , assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then R , the distance travelled, with dimension L, v x {\displaystyle v_{\text{x}}} , v y {\displaystyle v_{\text{y}}} , both dimensioned as T −1 L, and g the downward acceleration of gravity, with dimension T −2 L.
With these four quantities, we may conclude that the equation for the range R may be written:
Or dimensionally
from which we may deduce that a + b + c = 1 {\displaystyle a+b+c=1} and a + b + 2 c = 0 {\displaystyle a+b+2c=0} , which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.
However, if we use directed length dimensions, then v x {\displaystyle v_{\mathrm {x} }} will be dimensioned as T −1 L x , v y {\displaystyle v_{\mathrm {y} }} as T −1 L y , R as L x and g as T −2 L y . The dimensional equation becomes:
and we may solve completely as a = 1 , b = 1 and c = −1 . The increase in deductive power gained by the use of directed length dimensions is apparent.
Huntley's concept of directed length dimensions however has some serious limitations:
It also is often quite difficult to assign the L, L x , L y , L z , symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?
Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.
In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia ( inertial mass ), and mass as a measure of the quantity of matter. Quantity of matter is defined by Huntley as a quantity only proportional to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.
For example, consider the derivation of Poiseuille's Law . We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables:
There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:
If we distinguish between inertial mass with dimension M i {\displaystyle M_{\text{i}}} and quantity of matter with dimension M m {\displaystyle M_{\text{m}}} , then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:
where now only C is an undetermined constant (found to be equal to π / 8 {\displaystyle \pi /8} by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law .
Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements he postulated for it. For a given substance, the SI dimension amount of substance , with unit mole , does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.
Angles are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested [ 53 ] ) . As an example, consider again the projectile problem in which a point mass is launched from the origin ( x , y ) = (0, 0) at a speed v and angle θ above the x -axis, with the force of gravity directed along the negative y -axis. It is desired to find the range R , at which point the mass returns to the x -axis. Conventional analysis will yield the dimensionless variable π = R g / v 2 , but offers no insight into the relationship between R and θ .
Siano has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1 x 1 y 1 z to denote vector directions, and an orientationless symbol 1 0 . [ 54 ] Thus, Huntley's L x becomes L1 x with L specifying the dimension of length, and 1 x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1 i −1 = 1 i , the following multiplication table for the orientation symbols results:
The orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of 1 z . For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1 x and the side opposite has an orientation 1 y . Since (using ~ to indicate orientational equivalence) tan( θ ) = θ + ... ~ 1 y /1 x we conclude that an angle in the xy-plane must have an orientation 1 y /1 x = 1 z , which is not unreasonable. Analogous reasoning forces the conclusion that sin( θ ) has orientation 1 z while cos( θ ) has orientation 1 0 . These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos( θ ) + b sin( θ ) , where a and b are real scalars. An expression such as sin ( θ + π / 2 ) = cos ( θ ) {\displaystyle \sin(\theta +\pi /2)=\cos(\theta )} is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:
which for a = θ {\displaystyle a=\theta } and b = π / 2 {\displaystyle b=\pi /2} yields sin ( θ 1 z + [ π / 2 ] 1 z ) = 1 z cos ( θ 1 z ) {\displaystyle \sin(\theta \,1_{\text{z}}+[\pi /2]\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})} . Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is 1 0 {\displaystyle 1_{0}} .
The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral, putting it into normal form . The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd.
As an example, for the projectile problem, using orientational symbols, θ , being in the xy-plane will thus have dimension 1 z and the range of the projectile R will be of the form:
Dimensional homogeneity will now correctly yield a = −1 and b = 2 , and orientational homogeneity requires that 1 x / ( 1 y a 1 z c ) = 1 z c + 1 = 1 {\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1} . In other words, that c must be an odd integer. In fact, the required function of theta will be sin( θ )cos( θ ) which is a series consisting of odd powers of θ .
It is seen that the Taylor series of sin( θ ) and cos( θ ) are orientationally homogeneous using the above multiplication table, while expressions like cos( θ ) + sin( θ ) and exp( θ ) are not, and are (correctly) deemed unphysical.
Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the radian may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis. | https://en.wikipedia.org/wiki/Quantity_equation |
In the construction industry , a quantity surveyor ( QS ) is a professional with expert knowledge of construction costs and contracting . Qualified professional quantity surveyors can be known as Chartered Surveyors (Members and Fellows of RICS ) in the UK and Certified Quantity Surveyors (a designation of the Australian Institute of Quantity Surveyors) in Australia and other countries. In some countries, including Canada, South Africa, Kenya and Mauritius, qualified quantity surveyors are known as Professional Quantity Surveyors, a title protected by law. [ 1 ] [ 2 ] [ 3 ] Due to a shift in the construction industry and the increased demand for Quantity Surveying expertise, today less importance is being placed on Charterships, with a large percentage of working Quantity Surveyors practising with College / University degrees and without membership or fellowship to professional associations.
Quantity surveyors are responsible for managing all aspects of the contractual and financial side of construction projects. They help to ensure that the construction project is completed within its projected budget. Quantity surveyors are also hired by contractors to help with the valuation of construction work for the contractor, help with bidding and project budgeting, and the submission of bills to the client.
The duties of a quantity surveyor are as follows:
A university degree or diploma alone does not allow one to register as a Chartered Quantity Surveyor. Usually, anyone looking to qualify as a Chartered Quantity Surveyor, Certified Quantity Surveyor must hold appropriate educational qualifications and work experience, and must pass a professional competence assessment.
The RICS requires an RICS approved degree, several years of practical experience, and passing the Assessment of Professional Competence (APC) to qualify as a Chartered Quantity Surveyor. Some candidates may be entitled to qualify through extensive experience and reciprocity agreements. [ 4 ]
As construction projects become increasingly complex, the demand for skilled quantity surveyors continues to grow. The importance of Quantity Surveyors becoming Chartered is lessening year on year, with more and more businesses opting to hire staff with a standard Quantity Surveying degree and develop Quantity Surveying skills through their own training programmes. The future of quantity surveying lies in embracing digitalization, automation, and sustainable practices. Quantity surveyors will play a pivotal role in managing costs, optimizing resources, and ensuring the financial success of construction projects. [ 5 ] | https://en.wikipedia.org/wiki/Quantity_surveyor |
Quantity take-offs (QTO) are a detailed measurement of materials and labor needed to complete a construction project. They are developed by an estimator during the pre-construction phase. This process includes breaking the project down into smaller and more manageable units that are easier to measure or estimate. The level of detail required for measurement may vary. [ 1 ] These measurements are used to format a bid on the scope of construction . Estimators review drawings , specifications and models to find these quantities. Experienced estimators have developed procedures to help them quantify their work. Many programs have been developed to aid in the efficiency of these processes. With BIM quantity take-off can be conducted almost automatically given that the type of materials, their quantity and price is included in the model. [ 2 ] It is known that construction projects often run overtime and over budget and one of the reasons is lack of accuracy in quantity takeoff and estimates. [ 3 ] [ 2 ]
For every project's success accurate quantity takeoff is necessary. It minimizes many risks during the project. By leveraging precise quantity data, professionals can streamline project management, maintain budgetary control, and foster collaboration among stakeholders. [ 4 ]
This labor -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantity_take-off |
A physical quantity (or simply quantity ) [ 1 ] [ a ] is a property of a material or system that can be quantified by measurement . A physical quantity can be expressed as a value , which is the algebraic multiplication of a numerical value and a unit of measurement . For example, the physical quantity mass , symbol m , can be quantified as m = n kg, where n is the numerical value and kg is the unit symbol (for kilogram ). Quantities that are vectors have, besides numerical value and unit, direction or orientation in space.
Following ISO 80000-1 , [ 1 ] any value or magnitude of a physical quantity is expressed as a comparison to a unit of that quantity. The value of a physical quantity Z is expressed as the product of a numerical value { Z } (a pure number) and a unit [ Z ]:
For example, let Z {\displaystyle Z} be "2 metres"; then, { Z } = 2 {\displaystyle \{Z\}=2} is the numerical value and [ Z ] = m e t r e {\displaystyle [Z]=\mathrm {metre} } is the unit.
Conversely, the numerical value expressed in an arbitrary unit can be obtained as:
The multiplication sign is usually left out, just as it is left out between variables in the scientific notation of formulas. The convention used to express quantities is referred to as quantity calculus . In formulas, the unit [ Z ] can be treated as if it were a specific magnitude of a kind of physical dimension : see Dimensional analysis for more on this treatment.
International recommendations for the use of symbols for quantities are set out in ISO/IEC 80000 , the IUPAP red book and the IUPAC green book . For example, the recommended symbol for the physical quantity "mass" is m , and the recommended symbol for the quantity "electric charge" is Q .
Physical quantities are normally typeset in italics.
Purely numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes in italics. Symbols for elementary functions (circular trigonometric, hyperbolic, logarithmic etc.), changes in a quantity like Δ in Δ y or operators like d in d x , are also recommended to be printed in roman type.
Examples:
A scalar is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of the Latin or Greek alphabet , and are printed in italic type.
Vectors are physical quantities that possess both magnitude and direction and whose operations obey the axioms of a vector space . Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. For example, if u is the speed of a particle, then the straightforward notations for its velocity are u , u , or u → {\displaystyle {\vec {u}}} .
Scalar and vector quantities are the simplest tensor quantities , which are tensors that can be used to describe more general physical properties. For example, the Cauchy stress tensor possesses magnitude, direction, and orientation qualities.
The notion of dimension of a physical quantity was introduced by Joseph Fourier in 1822. [ 2 ] By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension.
There is often a choice of unit, though SI units are usually used in scientific contexts due to their ease of use, international familiarity and prescription. For example, a quantity of mass might be represented by the symbol m , and could be expressed in the units kilograms (kg), pounds (lb), or daltons (Da).
Dimensional homogeneity is not necessarily sufficient for quantities to be comparable; [ 1 ] for example, both kinematic viscosity and thermal diffusivity have dimension of square length per time (in units of m 2 /s ).
Quantities of the same kind share extra commonalities beyond their dimension and units allowing their comparison;
for example, not all dimensionless quantities are of the same kind. [ 1 ]
A systems of quantities relates physical quantities, and due to this dependence, a limited number of quantities can serve as a basis in terms of which the dimensions of all the remaining quantities of the system can be defined. A set of mutually independent quantities may be chosen by convention to act as such a set, and are called base quantities. The seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units and dimensions are listed in the following table. [ 3 ] : 136 Other conventions may have a different number of base units (e.g. the CGS and MKS systems of units).
The angular quantities, plane angle and solid angle , are defined as derived dimensionless quantities in the SI. For some relations, their units radian and steradian can be written explicitly to emphasize the fact that the quantity involves plane or solid angles. [ 3 ] : 137
Derived quantities are those whose definitions are based on other physical quantities (base quantities).
Important applied base units for space and time are below. Area and volume are thus, of course, derived from the length, but included for completeness as they occur frequently in many derived quantities, in particular densities.
Important and convenient derived quantities such as densities, fluxes , flows , currents are associated with many quantities. Sometimes different terms such as current density and flux density , rate , frequency and current , are used interchangeably in the same context; sometimes they are used uniquely.
To clarify these effective template-derived quantities, we use q to stand for any quantity within some scope of context (not necessarily base quantities) and present in the table below some of the most commonly used symbols where applicable, their definitions, usage, SI units and SI dimensions – where [ q ] denotes the dimension of q .
For time derivatives, specific, molar, and flux densities of quantities, there is no one symbol; nomenclature depends on the subject, though time derivatives can be generally written using overdot notation. For generality we use q m , q n , and F respectively. No symbol is necessarily required for the gradient of a scalar field, since only the nabla/del operator ∇ or grad needs to be written. For spatial density, current, current density and flux, the notations are common from one context to another, differing only by a change in subscripts.
For current density, t ^ {\displaystyle \mathbf {\hat {t}} } is a unit vector in the direction of flow, i.e. tangent to a flowline. Notice the dot product with the unit normal for a surface, since the amount of current passing through the surface is reduced when the current is not normal to the area. Only the current passing perpendicular to the surface contributes to the current passing through the surface, no current passes in the (tangential) plane of the surface.
The calculus notations below can be used synonymously.
If X is a n -variable function X ≡ X ( x 1 , x 2 ⋯ x n ) {\displaystyle X\equiv X\left(x_{1},x_{2}\cdots x_{n}\right)} , then
Differential The differential n -space volume element is d n x ≡ d V n ≡ d x 1 d x 2 ⋯ d x n {\displaystyle \mathrm {d} ^{n}x\equiv \mathrm {d} V_{n}\equiv \mathrm {d} x_{1}\mathrm {d} x_{2}\cdots \mathrm {d} x_{n}} ,
No common symbol for n -space density, here ρ n is used.
(length, area, volume or higher dimensions)
q = ∫ q λ d λ {\displaystyle q=\int q_{\lambda }\mathrm {d} \lambda } q = ∫ q ν d ν {\displaystyle q=\int q_{\nu }\mathrm {d} \nu }
[q]T ( q ν )
Transport mechanics , nuclear physics / particle physics : q = ∭ F d A d t {\displaystyle q=\iiint F\mathrm {d} A\mathrm {d} t}
Vector field : Φ F = ∬ S F ⋅ d A {\displaystyle \Phi _{F}=\iint _{S}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }
k -vector q : m = r ∧ q {\displaystyle \mathbf {m} =\mathbf {r} \wedge q} | https://en.wikipedia.org/wiki/Quantity_value |
Quantization , involved in image processing , is a lossy compression technique achieved by compressing a range of values to a single quantum (discrete) value. When the number of discrete symbols in a given stream is reduced, the stream becomes more compressible. For example, reducing the number of colors required to represent a digital image makes it possible to reduce its file size. Specific applications include DCT data quantization in JPEG and DWT data quantization in JPEG 2000 .
Color quantization reduces the number of colors used in an image; this is important for displaying images on devices that support a limited number of colors and for efficiently compressing certain kinds of images. Most bitmap editors and many operating systems have built-in support for color quantization. Popular modern color quantization algorithms include the nearest color algorithm (for fixed palettes), the median cut algorithm , and an algorithm based on octrees .
It is common to combine color quantization with dithering to create an impression of a larger number of colors and eliminate banding artifacts.
Grayscale quantization, also known as gray level quantization, is a process in digital image processing that involves reducing the number of unique intensity levels (shades of gray) in an image while preserving its essential visual information. This technique is commonly used for simplifying images, reducing storage requirements, and facilitating processing operations. In grayscale quantization, an image with N intensity levels is converted into an image with a reduced number of levels, typically L levels, where L < N . The process involves mapping each pixel's original intensity value to one of the new intensity levels. One of the simplest methods of grayscale quantization is uniform quantization, where the intensity range is divided into equal intervals, and each interval is represented by a single intensity value. Let's say we have an image with intensity levels ranging from 0 to 255 (8-bit grayscale). If we want to quantize it to 4 levels, the intervals would be [0-63], [64-127], [128-191], and [192-255]. Each interval would be represented by the midpoint intensity value, resulting in intensity levels of 31, 95, 159, and 223 respectively.
The formula for uniform quantization is:
Q ( x ) = ⌊ x Δ ⌋ × Δ + Δ 2 {\displaystyle Q(x)=\left\lfloor {\frac {x}{\Delta }}\right\rfloor \times \Delta +{\frac {\Delta }{2}}} Where:
Let's quantize an original intensity value of 147 to 3 intensity levels.
Original intensity value: x =147
Desired intensity levels: L =3
We first need to calculate the size of each quantization interval:
Δ = 255 L − 1 = 255 3 − 1 = 127.5 {\displaystyle \Delta ={\frac {255}{L-1}}={\frac {255}{3-1}}=127.5}
Using the uniform quantization formula:
Q ( x ) = ⌊ 147 127.5 ⌋ × 127.5 + 127.5 2 {\displaystyle Q(x)=\left\lfloor {\frac {147}{127.5}}\right\rfloor \times 127.5+{\frac {127.5}{2}}}
Q ( x ) = ⌊ 1.15294118 ⌋ × 127.5 + 127.5 2 {\displaystyle Q(x)=\left\lfloor 1.15294118\right\rfloor \times 127.5+{\frac {127.5}{2}}}
Q ( x ) = 1 × 127.5 + 63.75 = 191.25 {\displaystyle Q(x)=1\times 127.5+63.75=191.25}
Rounding 191.25 to the nearest integer, we get Q ( x ) = 191 {\displaystyle Q(x)=191}
So, the quantized intensity value of 147 to 3 levels is 191.
The human eye is fairly good at seeing small differences in brightness over a relatively large area, but not so good at distinguishing the exact strength of a high frequency (rapidly varying) brightness variation. This fact allows one to reduce the amount of information required by ignoring the high frequency components. This is done by simply dividing each component in the frequency domain by a constant for that component, and then rounding to the nearest integer. This is the main lossy operation in the whole process. As a result of this, it is typically the case that many of the higher frequency components are rounded to zero, and many of the rest become small positive or negative numbers.
As human vision is also more sensitive to luminance than chrominance , further compression can be obtained by working in a non-RGB color space which separates the two (e.g., YCbCr ), and quantizing the channels separately. [ 1 ]
A typical video codec works by breaking the picture into discrete blocks (8×8 pixels in the case of MPEG [ 1 ] ). These blocks can then be subjected to discrete cosine transform (DCT) to calculate the frequency components, both horizontally and vertically. [ 1 ] The resulting block (the same size as the original block) is then pre-multiplied by the quantization scale code and divided element-wise by the quantization matrix, and rounding each resultant element. The quantization matrix is designed to provide more resolution to more perceivable frequency components over less perceivable components (usually lower frequencies over high frequencies) in addition to transforming as many components to 0, which can be encoded with greatest efficiency. Many video encoders (such as DivX , Xvid , and 3ivx ) and compression standards (such as MPEG-2 and H.264/AVC ) allow custom matrices to be used. The extent of the reduction may be varied by changing the quantizer scale code, taking up much less bandwidth than a full quantizer matrix. [ 1 ]
This is an example of DCT coefficient matrix:
A common quantization matrix is:
Dividing the DCT coefficient matrix element-wise with this quantization matrix, and rounding to integers results in:
For example, using −415 (the DC coefficient) and rounding to the nearest integer
Typically this process will result in matrices with values primarily in the upper left (low frequency) corner. By using a zig-zag ordering to group the non-zero entries and run length encoding , the quantized matrix can be much more efficiently stored than the non-quantized version. [ 1 ]
[ 1 ] | https://en.wikipedia.org/wiki/Quantization_(image_processing) |
Quantization (in British English quantisation ) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics . It is a procedure for constructing quantum mechanics from classical mechanics . A generalization involving infinite degrees of freedom is field quantization , as in the "quantization of the electromagnetic field ", referring to photons as field " quanta " (for instance as light quanta ). This procedure is basic to theories of atomic physics , chemistry, particle physics , nuclear physics , condensed matter physics , and quantum optics .
In 1901, when Max Planck was developing the distribution function of statistical mechanics to solve the ultraviolet catastrophe problem, he realized that the properties of blackbody radiation can be explained by the assumption that the amount of energy must be in countable fundamental units, i.e. amount of energy is not continuous but discrete . That is, a minimum unit of energy exists and the following relationship holds E = h ν {\displaystyle E=h\nu } for the frequency ν {\displaystyle \nu } . Here, h {\displaystyle h} is called the Planck constant , which represents the amount of the quantum mechanical effect. It means a fundamental change of mathematical model of physical quantities.
In 1905, Albert Einstein published a paper, "On a heuristic viewpoint concerning the emission and transformation of light", which explained the photoelectric effect on quantized electromagnetic waves . [ 1 ] The energy quantum referred to in this paper was later called " photon ". In July 1913, Niels Bohr used quantization to describe the spectrum of a hydrogen atom in his paper "On the constitution of atoms and molecules".
The preceding theories have been successful, but they are very phenomenological theories. However, the French mathematician Henri Poincaré first gave a systematic and rigorous definition of what quantization is in his 1912 paper "Sur la théorie des quanta". [ 2 ] [ 3 ]
The term "quantum physics" was first used in Johnston's Planck's Universe in Light of Modern Physics . (1931).
Canonical quantization develops quantum mechanics from classical mechanics . One introduces a commutation relation among canonical coordinates . Technically, one converts coordinates to operators, through combinations of creation and annihilation operators . The operators act on quantum states of the theory. The lowest energy state is called the vacuum state .
Even within the setting of canonical quantization, there is difficulty associated to quantizing arbitrary observables on the classical phase space. This is the ordering ambiguity : classically, the position and momentum variables x and p commute, but their quantum mechanical operator counterparts do not. Various quantization schemes have been proposed to resolve this ambiguity, [ 4 ] of which the most popular is the Weyl quantization scheme . Nevertheless, Groenewold's theorem dictates that no perfect quantization scheme exists. Specifically, if the quantizations of x and p are taken to be the usual position and momentum operators, then no quantization scheme can perfectly reproduce the Poisson bracket relations among the classical observables. [ 5 ]
There is a way to perform a canonical quantization without having to resort to the non covariant approach of foliating spacetime and choosing a Hamiltonian . This method is based upon a classical action, but is different from the functional integral approach.
The method does not apply to all possible actions (for instance, actions with a noncausal structure or actions with gauge "flows" ). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by the Euler–Lagrange equations . Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket . This Poisson algebra is then ℏ -deformed in the same way as in canonical quantization.
In quantum field theory , there is also a way to quantize actions with gauge "flows" . It involves the Batalin–Vilkovisky formalism , an extension of the BRST formalism .
One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. [ 6 ] Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold [ 7 ] considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions.
More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor ), Weyl's map is not satisfactory.
For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ 2 / 2 . (This extra term offset is pedagogically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom, even though the standard QM ground state of the atom has vanishing l .) [ 8 ]
As a mere representation change , however, Weyl's map is useful and important, as it underlies the alternate equivalent phase space formulation of conventional quantum mechanics.
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.
A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s by Bertram Kostant and Jean-Marie Souriau . The method proceeds in two stages. [ 9 ] First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket relations. On the other hand, this prequantum Hilbert space is too big to be physically meaningful. One then restricts to functions (or sections) depending on half the variables on the phase space, yielding the quantum Hilbert space.
A classical mechanical theory is given by an action with the permissible configurations being the ones which are extremal with respect to functional variations of the action. A quantum-mechanical description of the classical system can also be constructed from the action of the system by means of the path integral formulation . | https://en.wikipedia.org/wiki/Quantization_(physics) |
Quantization , in mathematics and digital signal processing , is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements . Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms.
The difference between an input value and its quantized value (such as round-off error ) is referred to as quantization error , noise or distortion . A device or algorithmic function that performs quantization is called a quantizer . An analog-to-digital converter is an example of a quantizer.
For example, rounding a real number x {\displaystyle x} to the nearest integer value forms a very basic type of quantizer – a uniform one. A typical ( mid-tread ) uniform quantizer with a quantization step size equal to some value Δ {\displaystyle \Delta } can be expressed as
where the notation ⌊ ⌋ {\displaystyle \lfloor \ \rfloor } denotes the floor function .
Alternatively, the same quantizer may be expressed in terms of the ceiling function , as
(The notation ⌈ ⌉ {\displaystyle \lceil \ \rceil } denotes the ceiling function).
The essential property of a quantizer is having a countable set of possible output values smaller than the set of possible input values. The members of the set of output values may have integer, rational, or real values. For simple rounding to the nearest integer, the step size Δ {\displaystyle \Delta } is equal to 1. With Δ = 1 {\displaystyle \Delta =1} or with Δ {\displaystyle \Delta } equal to any other integer value, this quantizer has real-valued inputs and integer-valued outputs.
When the quantization step size (Δ) is small relative to the variation in the signal being quantized, it is relatively simple to show that the mean squared error produced by such a rounding operation will be approximately Δ 2 / 12 {\displaystyle \Delta ^{2}/12} . [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Mean squared error is also called the quantization noise power . Adding one bit to the quantizer halves the value of Δ, which reduces the noise power by the factor 1 / 4 . In terms of decibels , the noise power change is 10 ⋅ log 10 ( 1 / 4 ) ≈ − 6 d B . {\displaystyle \scriptstyle 10\cdot \log _{10}(1/4)\ \approx \ -6\ \mathrm {dB} .}
Because the set of possible output values of a quantizer is countable, any quantizer can be decomposed into two distinct stages, which can be referred to as the classification stage (or forward quantization stage) and the reconstruction stage (or inverse quantization stage), where the classification stage maps the input value to an integer quantization index k {\displaystyle k} and the reconstruction stage maps the index k {\displaystyle k} to the reconstruction value y k {\displaystyle y_{k}} that is the output approximation of the input value. For the example uniform quantizer described above, the forward quantization stage can be expressed as
and the reconstruction stage for this example quantizer is simply
This decomposition is useful for the design and analysis of quantization behavior, and it illustrates how the quantized data can be communicated over a communication channel – a source encoder can perform the forward quantization stage and send the index information through a communication channel, and a decoder can perform the reconstruction stage to produce the output approximation of the original input data. In general, the forward quantization stage may use any function that maps the input data to the integer space of the quantization index data, and the inverse quantization stage can conceptually (or literally) be a table look-up operation to map each quantization index to a corresponding reconstruction value. This two-stage decomposition applies equally well to vector as well as scalar quantizers.
Because quantization is a many-to-few mapping, it is an inherently non-linear and irreversible process (i.e., because the same output value is shared by multiple input values, it is impossible, in general, to recover the exact input value when given only the output value).
The set of possible input values may be infinitely large, and may possibly be continuous and therefore uncountable (such as the set of all real numbers, or all real numbers within some limited range). The set of possible output values may be finite or countably infinite . [ 6 ] The input and output sets involved in quantization can be defined in a rather general way. For example, vector quantization is the application of quantization to multi-dimensional (vector-valued) input data. [ 7 ]
An analog-to-digital converter (ADC) can be modeled as two processes: sampling and quantization. Sampling converts a time-varying voltage signal into a discrete-time signal , a sequence of real numbers. Quantization replaces each real number with an approximation from a finite set of discrete values. Most commonly, these discrete values are represented as fixed-point words. Though any number of quantization levels is possible, common word lengths are 8-bit (256 levels), 16-bit (65,536 levels) and 24-bit (16.8 million levels). Quantizing a sequence of numbers produces a sequence of quantization errors which is sometimes modeled as an additive random signal called quantization noise because of its stochastic behavior. The more levels a quantizer uses, the lower is its quantization noise power.
Rate–distortion optimized quantization is encountered in source coding for lossy data compression algorithms, where the purpose is to manage distortion within the limits of the bit rate supported by a communication channel or storage medium. The analysis of quantization in this context involves studying the amount of data (typically measured in digits or bits or bit rate ) that is used to represent the output of the quantizer and studying the loss of precision that is introduced by the quantization process (which is referred to as the distortion ).
Most uniform quantizers for signed input data can be classified as being of one of two types: mid-riser and mid-tread . The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a stairway . Mid-tread quantizers have a zero-valued reconstruction level (corresponding to a tread of a stairway), while mid-riser quantizers have a zero-valued classification threshold (corresponding to a riser of a stairway). [ 9 ]
Mid-tread quantization involves rounding. The formulas for mid-tread uniform quantization are provided in the previous section.
Mid-riser quantization involves truncation. The input-output formula for a mid-riser uniform quantizer is given by:
where the classification rule is given by
and the reconstruction rule is
Note that mid-riser uniform quantizers do not have a zero output value – their minimum output magnitude is half the step size. In contrast, mid-tread quantizers do have a zero output level. For some applications, having a zero output signal representation may be a necessity.
In general, a mid-riser or mid-tread quantizer may not actually be a uniform quantizer – i.e., the size of the quantizer's classification intervals may not all be the same, or the spacing between its possible output values may not all be the same. The distinguishing characteristic of a mid-riser quantizer is that it has a classification threshold value that is exactly zero, and the distinguishing characteristic of a mid-tread quantizer is that is it has a reconstruction value that is exactly zero. [ 9 ]
A dead-zone quantizer is a type of mid-tread quantizer with symmetric behavior around 0. The region around the zero output value of such a quantizer is referred to as the dead zone or deadband . The dead zone can sometimes serve the same purpose as a noise gate or squelch function. Especially for compression applications, the dead-zone may be given a different width than that for the other steps. For an otherwise-uniform quantizer, the dead-zone width can be set to any value w {\displaystyle w} by using the forward quantization rule [ 10 ] [ 11 ] [ 12 ]
where the function sgn {\displaystyle \operatorname {sgn} } ( ) is the sign function (also known as the signum function). The general reconstruction rule for such a dead-zone quantizer is given by
where r k {\displaystyle r_{k}} is a reconstruction offset value in the range of 0 to 1 as a fraction of the step size. Ordinarily, 0 ≤ r k ≤ 1 2 {\displaystyle 0\leq r_{k}\leq {\tfrac {1}{2}}} when quantizing input data with a typical probability density function (PDF) that is symmetric around zero and reaches its peak value at zero (such as a Gaussian , Laplacian , or generalized Gaussian PDF). Although r k {\displaystyle r_{k}} may depend on k {\displaystyle k} in general and can be chosen to fulfill the optimality condition described below, it is often simply set to a constant, such as 1 2 {\displaystyle {\tfrac {1}{2}}} . (Note that in this definition, y 0 = 0 {\displaystyle y_{0}=0} due to the definition of the sgn {\displaystyle \operatorname {sgn} } ( ) function, so r 0 {\displaystyle r_{0}} has no effect.)
A very commonly used special case (e.g., the scheme typically used in financial accounting and elementary mathematics) is to set w = Δ {\displaystyle w=\Delta } and r k = 1 2 {\displaystyle r_{k}={\tfrac {1}{2}}} for all k {\displaystyle k} . In this case, the dead-zone quantizer is also a uniform quantizer, since the central dead-zone of this quantizer has the same width as all of its other steps, and all of its reconstruction values are equally spaced as well.
A common assumption for the analysis of quantization error is that it affects a signal processing system in a similar manner to that of additive white noise – having negligible correlation with the signal and an approximately flat power spectral density . [ 2 ] [ 6 ] [ 13 ] [ 14 ] The additive noise model is commonly used for the analysis of quantization error effects in digital filtering systems, and it can be very useful in such analysis. It has been shown to be a valid model in cases of high-resolution quantization (small Δ {\displaystyle \Delta } relative to the signal strength) with smooth PDFs. [ 2 ] [ 15 ]
Additive noise behavior is not always a valid assumption. Quantization error (for quantizers defined as described here) is deterministically related to the signal and not entirely independent of it. Thus, periodic signals can create periodic quantization noise. And in some cases, it can even cause limit cycles to appear in digital signal processing systems. One way to ensure effective independence of the quantization error from the source signal is to perform dithered quantization (sometimes with noise shaping ), which involves adding random (or pseudo-random ) noise to the signal prior to quantization. [ 6 ] [ 14 ]
In the typical case, the original signal is much larger than one least significant bit (LSB). When this is the case, the quantization error is not significantly correlated with the signal and has an approximately uniform distribution . When rounding is used to quantize, the quantization error has a mean of zero and the root mean square (RMS) value is the standard deviation of this distribution, given by 1 12 L S B ≈ 0.289 L S B {\displaystyle \scriptstyle {\frac {1}{\sqrt {12}}}\mathrm {LSB} \ \approx \ 0.289\,\mathrm {LSB} } . When truncation is used, the error has a non-zero mean of 1 2 L S B {\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} } and the RMS value is 1 3 L S B {\displaystyle \scriptstyle {\frac {1}{\sqrt {3}}}\mathrm {LSB} } . Although rounding yields less RMS error than truncation, the difference is only due to the static (DC) term of 1 2 L S B {\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} } . The RMS values of the AC error are exactly the same in both cases, so there is no special advantage of rounding over truncation in situations where the DC term of the error can be ignored (such as in AC-coupled systems). In either case, the standard deviation, as a percentage of the full signal range, changes by a factor of 2 for each 1-bit change in the number of quantization bits. The potential signal-to-quantization-noise power ratio therefore changes by 4, or 10 ⋅ log 10 ( 4 ) {\displaystyle \scriptstyle 10\cdot \log _{10}(4)} , approximately 6 dB per bit.
At lower amplitudes the quantization error becomes dependent on the input signal, resulting in distortion. This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the band of interest. In order to make the quantization error independent of the input signal, the signal is dithered by adding noise to the signal. This slightly reduces signal-to-noise ratio, but can completely eliminate the distortion.
Quantization noise is a model of quantization error introduced by quantization in the ADC. It is a rounding error between the analog input voltage to the ADC and the output digitized value. The noise is non-linear and signal-dependent. It can be modeled in several different ways.
In an ideal ADC, where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the Signal-to-quantization-noise ratio (SQNR) can be calculated from
where Q is the number of quantization bits.
The most common test signals that fulfill this are full amplitude triangle waves and sawtooth waves .
For example, a 16-bit ADC has a maximum signal-to-quantization-noise ratio of 6.02 × 16 = 96.3 dB.
When the input signal is a full-amplitude sine wave the distribution of the signal is no longer uniform, and the corresponding equation is instead
Here, the quantization noise is once again assumed to be uniformly distributed. When the input signal has a high amplitude and a wide frequency spectrum this is the case. [ 16 ] In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09 dB. The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle or sawtooth.
For complex signals in high-resolution ADCs this is an accurate model. For low-resolution ADCs, low-level signals in high-resolution ADCs, and for simple waveforms the quantization noise is not uniformly distributed, making this model inaccurate. [ 17 ] In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.
The calculations are relative to full-scale input. For smaller signals, the relative quantization distortion can be very large. To circumvent this issue, analog companding can be used, but this can introduce distortion.
Often the design of a quantizer involves supporting only a limited range of possible output values and performing clipping to limit the output to this range whenever the input exceeds the supported range. The error introduced by this clipping is referred to as overload distortion. Within the extreme limits of the supported range, the amount of spacing between the selectable output values of a quantizer is referred to as its granularity , and the error introduced by this spacing is referred to as granular distortion. It is common for the design of a quantizer to involve determining the proper balance between granular distortion and overload distortion. For a given supported number of possible output values, reducing the average granular distortion may involve increasing the average overload distortion, and vice versa. A technique for controlling the amplitude of the signal (or, equivalently, the quantization step size Δ {\displaystyle \Delta } ) to achieve the appropriate balance is the use of automatic gain control (AGC). However, in some quantizer designs, the concepts of granular error and overload error may not apply (e.g., for a quantizer with a limited range of input data or with a countably infinite set of selectable output values). [ 6 ]
A scalar quantizer, which performs a quantization operation, can ordinarily be decomposed into two stages:
These two stages together comprise the mathematical operation of y = Q ( x ) {\displaystyle y=Q(x)} .
Entropy coding techniques can be applied to communicate the quantization indices from a source encoder that performs the classification stage to a decoder that performs the reconstruction stage. One way to do this is to associate each quantization index k {\displaystyle k} with a binary codeword c k {\displaystyle c_{k}} . An important consideration is the number of bits used for each codeword, denoted here by l e n g t h ( c k ) {\displaystyle \mathrm {length} (c_{k})} . As a result, the design of an M {\displaystyle M} -level quantizer and an associated set of codewords for communicating its index values requires finding the values of { b k } k = 1 M − 1 {\displaystyle \{b_{k}\}_{k=1}^{M-1}} , { c k } k = 1 M {\displaystyle \{c_{k}\}_{k=1}^{M}} and { y k } k = 1 M {\displaystyle \{y_{k}\}_{k=1}^{M}} which optimally satisfy a selected set of design constraints such as the bit rate R {\displaystyle R} and distortion D {\displaystyle D} .
Assuming that an information source S {\displaystyle S} produces random variables X {\displaystyle X} with an associated PDF f ( x ) {\displaystyle f(x)} , the probability p k {\displaystyle p_{k}} that the random variable falls within a particular quantization interval I k {\displaystyle I_{k}} is given by:
The resulting bit rate R {\displaystyle R} , in units of average bits per quantized value, for this quantizer can be derived as follows:
If it is assumed that distortion is measured by mean squared error, [ a ] the distortion D , is given by:
A key observation is that rate R {\displaystyle R} depends on the decision boundaries { b k } k = 1 M − 1 {\displaystyle \{b_{k}\}_{k=1}^{M-1}} and the codeword lengths { l e n g t h ( c k ) } k = 1 M {\displaystyle \{\mathrm {length} (c_{k})\}_{k=1}^{M}} , whereas the distortion D {\displaystyle D} depends on the decision boundaries { b k } k = 1 M − 1 {\displaystyle \{b_{k}\}_{k=1}^{M-1}} and the reconstruction levels { y k } k = 1 M {\displaystyle \{y_{k}\}_{k=1}^{M}} .
After defining these two performance metrics for the quantizer, a typical rate–distortion formulation for a quantizer design problem can be expressed in one of two ways:
Often the solution to these problems can be equivalently (or approximately) expressed and solved by converting the formulation to the unconstrained problem min { D + λ ⋅ R } {\displaystyle \min \left\{D+\lambda \cdot R\right\}} where the Lagrange multiplier λ {\displaystyle \lambda } is a non-negative constant that establishes the appropriate balance between rate and distortion. Solving the unconstrained problem is equivalent to finding a point on the convex hull of the family of solutions to an equivalent constrained formulation of the problem. However, finding a solution – especially a closed-form solution – to any of these three problem formulations can be difficult. Solutions that do not require multi-dimensional iterative optimization techniques have been published for only three PDFs: the uniform, [ 18 ] exponential , [ 12 ] and Laplacian [ 12 ] distributions. Iterative optimization approaches can be used to find solutions in other cases. [ 6 ] [ 19 ] [ 20 ]
Note that the reconstruction values { y k } k = 1 M {\displaystyle \{y_{k}\}_{k=1}^{M}} affect only the distortion – they do not affect the bit rate – and that each individual y k {\displaystyle y_{k}} makes a separate contribution d k {\displaystyle d_{k}} to the total distortion as shown below:
where
This observation can be used to ease the analysis – given the set of { b k } k = 1 M − 1 {\displaystyle \{b_{k}\}_{k=1}^{M-1}} values, the value of each y k {\displaystyle y_{k}} can be optimized separately to minimize its contribution to the distortion D {\displaystyle D} .
For the mean-square error distortion criterion, it can be easily shown that the optimal set of reconstruction values { y k ∗ } k = 1 M {\displaystyle \{y_{k}^{*}\}_{k=1}^{M}} is given by setting the reconstruction value y k {\displaystyle y_{k}} within each interval I k {\displaystyle I_{k}} to the conditional expected value (also referred to as the centroid ) within the interval, as given by:
The use of sufficiently well-designed entropy coding techniques can result in the use of a bit rate that is close to the true information content of the indices { k } k = 1 M {\displaystyle \{k\}_{k=1}^{M}} , such that effectively
and therefore
The use of this approximation can allow the entropy coding design problem to be separated from the design of the quantizer itself. Modern entropy coding techniques such as arithmetic coding can achieve bit rates that are very close to the true entropy of a source, given a set of known (or adaptively estimated) probabilities { p k } k = 1 M {\displaystyle \{p_{k}\}_{k=1}^{M}} .
In some designs, rather than optimizing for a particular number of classification regions M {\displaystyle M} , the quantizer design problem may include optimization of the value of M {\displaystyle M} as well. For some probabilistic source models, the best performance may be achieved when M {\displaystyle M} approaches infinity.
In the above formulation, if the bit rate constraint is neglected by setting λ {\displaystyle \lambda } equal to 0, or equivalently if it is assumed that a fixed-length code (FLC) will be used to represent the quantized data instead of a variable-length code (or some other entropy coding technology such as arithmetic coding that is better than an FLC in the rate–distortion sense), the optimization problem reduces to minimization of distortion D {\displaystyle D} alone.
The indices produced by an M {\displaystyle M} -level quantizer can be coded using a fixed-length code using R = ⌈ log 2 M ⌉ {\displaystyle R=\lceil \log _{2}M\rceil } bits/symbol. For example, when M = {\displaystyle M=} 256 levels, the FLC bit rate R {\displaystyle R} is 8 bits/symbol. For this reason, such a quantizer has sometimes been called an 8-bit quantizer. However using an FLC eliminates the compression improvement that can be obtained by use of better entropy coding.
Assuming an FLC with M {\displaystyle M} levels, the rate–distortion minimization problem can be reduced to distortion minimization alone. The reduced problem can be stated as follows: given a source X {\displaystyle X} with PDF f ( x ) {\displaystyle f(x)} and the constraint that the quantizer must use only M {\displaystyle M} classification regions, find the decision boundaries { b k } k = 1 M − 1 {\displaystyle \{b_{k}\}_{k=1}^{M-1}} and reconstruction levels { y k } k = 1 M {\displaystyle \{y_{k}\}_{k=1}^{M}} to minimize the resulting distortion
Finding an optimal solution to the above problem results in a quantizer sometimes called a MMSQE (minimum mean-square quantization error) solution, and the resulting PDF-optimized (non-uniform) quantizer is referred to as a Lloyd–Max quantizer, named after two people who independently developed iterative methods [ 6 ] [ 21 ] [ 22 ] to solve the two sets of simultaneous equations resulting from ∂ D / ∂ b k = 0 {\displaystyle {\partial D/\partial b_{k}}=0} and ∂ D / ∂ y k = 0 {\displaystyle {\partial D/\partial y_{k}}=0} , as follows:
which places each threshold at the midpoint between each pair of reconstruction values, and
which places each reconstruction value at the centroid (conditional expected value) of its associated classification interval.
Lloyd's Method I algorithm , originally described in 1957, can be generalized in a straightforward way for application to vector data. This generalization results in the Linde–Buzo–Gray (LBG) or k-means classifier optimization methods. Moreover, the technique can be further generalized in a straightforward way to also include an entropy constraint for vector data. [ 23 ]
The Lloyd–Max quantizer is actually a uniform quantizer when the input PDF is uniformly distributed over the range [ y 1 − Δ / 2 , y M + Δ / 2 ) {\displaystyle [y_{1}-\Delta /2,~y_{M}+\Delta /2)} . However, for a source that does not have a uniform distribution, the minimum-distortion quantizer may not be a uniform quantizer. The analysis of a uniform quantizer applied to a uniformly distributed source can be summarized in what follows:
A symmetric source X can be modelled with f ( x ) = 1 2 X max {\displaystyle f(x)={\tfrac {1}{2X_{\max }}}} , for x ∈ [ − X max , X max ] {\displaystyle x\in [-X_{\max },X_{\max }]} and 0 elsewhere.
The step size Δ = 2 X max M {\displaystyle \Delta ={\tfrac {2X_{\max }}{M}}} and the signal to quantization noise ratio (SQNR) of the quantizer is
For a fixed-length code using N {\displaystyle N} bits, M = 2 N {\displaystyle M=2^{N}} , resulting in S Q N R = 20 log 10 2 N = N ⋅ ( 20 log 10 2 ) = N ⋅ 6.0206 d B {\displaystyle {\rm {SQNR}}=20\log _{10}{2^{N}}=N\cdot (20\log _{10}2)=N\cdot 6.0206\,{\rm {dB}}} ,
or approximately 6 dB per bit. For example, for N {\displaystyle N} =8 bits, M {\displaystyle M} =256 levels and SQNR = 8×6 = 48 dB; and for N {\displaystyle N} =16 bits, M {\displaystyle M} =65536 and SQNR = 16×6 = 96 dB. The property of 6 dB improvement in SQNR for each extra bit used in quantization is a well-known figure of merit. However, it must be used with care: this derivation is only for a uniform quantizer applied to a uniform source. For other source PDFs and other quantizer designs, the SQNR may be somewhat different from that predicted by 6 dB/bit, depending on the type of PDF, the type of source, the type of quantizer, and the bit rate range of operation.
However, it is common to assume that for many sources, the slope of a quantizer SQNR function can be approximated as 6 dB/bit when operating at a sufficiently high bit rate. At asymptotically high bit rates, cutting the step size in half increases the bit rate by approximately 1 bit per sample (because 1 bit is needed to indicate whether the value is in the left or right half of the prior double-sized interval) and reduces the mean squared error by a factor of 4 (i.e., 6 dB) based on the Δ 2 / 12 {\displaystyle \Delta ^{2}/12} approximation.
At asymptotically high bit rates, the 6 dB/bit approximation is supported for many source PDFs by rigorous theoretical analysis. [ 2 ] [ 3 ] [ 5 ] [ 6 ] Moreover, the structure of the optimal scalar quantizer (in the rate–distortion sense) approaches that of a uniform quantizer under these conditions. [ 5 ] [ 6 ]
Many physical quantities are actually quantized by physical entities. Examples of fields where this limitation applies include electronics (due to electrons ), optics (due to photons ), biology (due to DNA ), physics (due to Planck limits ) and chemistry (due to molecules ). | https://en.wikipedia.org/wiki/Quantization_(signal_processing) |
In mathematics, more specifically in the context of geometric quantization , quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition [ 1 ] on the symplectic quotient of a compact symplectic manifold is the space of invariant sections [ vague ] of L .
This was conjectured in 1980s by Guillemin and Sternberg and was proven in 1990s by Meinrenken [ 2 ] [ 3 ] (the second paper used symplectic cut ) as well as Tian and Zhang. [ 4 ] For the formulation due to Teleman, see C. Woodward's notes.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantization_commutes_with_reduction |
In mathematics, a quantum or quantized enveloping algebra is a q -analog of a universal enveloping algebra . [ 1 ] Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} . The notation was introduced by Drinfeld and independently by Jimbo. [ 2 ]
Among the applications, studying the q → 0 {\displaystyle q\to 0} limit led to the discovery of crystal bases .
Michio Jimbo considered the algebras with three generators related by the three commutators
When η → 0 {\displaystyle \eta \to 0} , these reduce to the commutators that define the special linear Lie algebra s l 2 {\displaystyle {\mathfrak {sl}}_{2}} . In contrast, for nonzero η {\displaystyle \eta } , the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of s l 2 {\displaystyle {\mathfrak {sl}}_{2}} . [ 3 ]
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantized_enveloping_algebra |
In physics , a quantum ( pl. : quanta ) is the minimum amount of any physical entity ( physical property ) involved in an interaction . The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization ". [ 1 ] This means that the magnitude of the physical property can take on only discrete values consisting of integer multiples of one quantum. For example, a photon is a single quantum of light of a specific frequency (or of any other form of electromagnetic radiation ). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values. [ 2 ] Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom. Quantization is one of the foundations of the much broader physics of quantum mechanics . Quantization of energy and its influence on how energy and matter interact ( quantum electrodynamics ) is part of the fundamental framework for understanding and describing nature.
The modern concept of the quantum in physics originates from December 14, 1900, when Max Planck reported his findings to the German Physical Society . He showed that modelling harmonic oscillators with discrete energy levels resolved a longstanding problem in the theory of blackbody radiation . [ 3 ] : 15 [ 4 ] In his report, Planck did not use the term quantum in the modern sense. Instead, he used the term Elementarquantum to refer to the "quantum of electricity", now known as the elementary charge . For the smallest unit of energy, he employed the term Energieelement , "energy element", rather than calling it a quantum . [ 5 ]
Shortly afterwards, in a paper published in Annalen der Physik , [ 6 ] Planck introduced the constant h , which he termed the "quantum of action " ( elementares Wirkungsquantum ) in 1906. [ 5 ] In this paper, Planck also reported more precise values for the elementary charge and the Avogadro–Loschmidt number , the number of molecules in one mole of substance. [ 7 ] The constant h is now known as the Planck constant . After his theory was validated, Planck was awarded the Nobel Prize in Physics for his discovery in 1918. [ 8 ]
In 1905 Albert Einstein suggested that electromagnetic radiation exists in spatially localized packets which he called "quanta of light" ( Lichtquanta ). [ 5 ] [ 9 ] Einstein was able to use this hypothesis to recast Planck's treatment of the blackbody problem in a form that also explained the voltages observed in Philipp Lenard 's experiments on the photoelectric effect . [ 3 ] : 23 Shortly thereafter, the term "energy quantum" was introduced for the quantity hν . [ 10 ]
While quantization was first discovered in electromagnetic radiation , it describes a fundamental aspect of energy not just restricted to photons. [ 11 ] In the attempt to bring theory into agreement with experiment, Max Planck postulated that electromagnetic energy is absorbed or emitted in discrete packets, or quanta. [ 12 ] | https://en.wikipedia.org/wiki/Quantum |
The quantum-confined Stark effect ( QCSE ) describes the effect of an external electric field upon the light absorption spectrum or emission spectrum of a quantum well (QW). In the absence of an external electric field, electrons and holes within the quantum well may only occupy states within a discrete set of energy subbands. Only a discrete set of frequencies of light may be absorbed or emitted by the system. When an external electric field is applied, the electron states shift to lower energies, while the hole states shift to higher energies. This reduces the permitted light absorption or emission frequencies. Additionally, the external electric field shifts electrons and holes to opposite sides of the well, decreasing the overlap integral, which in turn reduces the recombination efficiency (i.e. fluorescence quantum yield ) of the system. [ 1 ] The spatial separation between the electrons and holes is limited by the presence of the potential barriers around the quantum well, meaning that excitons are able to exist in the system even under the influence of an electric field. The quantum-confined Stark effect is used in QCSE optical modulators , which allow optical communications signals to be switched on and off rapidly. [ 2 ]
Even if Quantum Objects (Wells, Dots or Discs, for instance) emit and absorb light generally with higher energies than the band gap of the material, the QCSE may shift the energy to values lower than the gap. This was evidenced recently in the study of quantum discs embedded in a nanowire. [ 3 ]
The shift in absorption lines can be calculated by comparing the energy levels in unbiased and biased quantum wells. It is a simpler task to find the energy levels in the unbiased system, due to its symmetry. If the external electric field is small, it can be treated as a perturbation to the unbiased system and its approximate effect can be found using perturbation theory .
The potential for a quantum well may be written as
where L {\displaystyle L} is the width of the well and V 0 {\displaystyle V_{0}} is the height of the potential barriers. The bound states in the well lie at a set of discrete energies, E n {\displaystyle E_{n}} and the associated wavefunctions can be written using the envelope function approximation as follows:
In this expression, A {\displaystyle A} is the cross-sectional area of the system, perpendicular to the quantization direction, u ( r ) {\displaystyle u(\mathbf {r} )} is a periodic Bloch function for the energy band edge in the bulk semiconductor and ϕ n ( z ) {\displaystyle \phi _{n}(z)} is a slowly varying envelope function for the system.
If the quantum well is very deep, it can be approximated by the particle in a box model, in which V 0 → ∞ {\displaystyle V_{0}\to \infty } . Under this simplified model, analytical expressions for the bound state wavefunctions exist, with the form
The energies of the bound states are
where m ∗ {\displaystyle m^{*}} is the effective mass of an electron in a given semiconductor.
Supposing the electric field is biased along the z direction,
the perturbing Hamiltonian term is
The first order correction to the energy levels is zero due to symmetry.
The second order correction is, for instance n=1,
for electron, where the additional approximation of neglecting the perturbation terms due to the bound states with k even and > 2 has been introduced. By comparison, the perturbation terms from odd-k states are zero due to symmetry.
Similar calculations can be applied to holes by replacing the electron effective mass m e ∗ {\displaystyle m_{e}^{*}} with the hole effective mass m h ∗ {\displaystyle m_{h}^{*}} . Introducing the total effective mass m t o t ∗ = m e ∗ + m h ∗ {\displaystyle m_{tot}^{*}=m_{e}^{*}+m_{h}^{*}} , the energy shift of the first optical transition induced by QCSE can be approximated to:
The approximations made so far are quite crude, nonetheless the energy shift does show experimentally a square law dependence from the applied electric field, [ 5 ] as predicted.
Additionally to the redshift towards lower energies of the optical transitions, the DC electric field also induces a decrease in magnitude of the absorption coefficient, as it decreases the overlapping integrals of relating valence and conduction band wave functions. Given the approximations made so far and the absence of any applied electric field along z, the overlapping integral for n v a l e n c e = n c o n d u c t i o n {\displaystyle n_{valence}=n_{conduction}} transitions will be:
To calculate how this integral is modified by the quantum-confined Stark effect we once again employ time independent perturbation theory .
The first order correction for the wave function is
Once again we look at the n = 1 {\displaystyle n=1} energy level and consider only the perturbation from the level n = 2 {\displaystyle n=2} (notice that the perturbation from n = 3 {\displaystyle n=3} would be = 0 {\displaystyle =0} due to symmetry). We obtain
for the conduction and valence band respectively, where A {\displaystyle A} has been introduced as a normalization constant. For any applied electric field F → ⋅ z ^ ≠ 0 {\displaystyle {\vec {F}}\cdot {\hat {z}}\neq 0} we obtain
Thus, according to Fermi's golden rule , which says that transition probability depends on the above overlapping integral, optical transition strength is weakened.
The description of quantum-confined Stark effect given by second order perturbation theory is extremely simple and intuitive. However to correctly depict QCSE the role of excitons has to be taken into account. Excitons are quasiparticles consisting of a bound state of an electron-hole pair, whose binding energy in a bulk material can be modelled as that of an hydrogenic atom
where R H {\displaystyle R_{H}} is the Rydberg constant , μ {\displaystyle \mu } is the reduced mass of the electron-hole pair and ε r {\displaystyle \varepsilon _{r}} is the relative electric permittivity.
The exciton binding energy has to be included in the energy balance of photon absorption processes:
Exciton generation therefore redshift the optical band gap towards lower energies.
If an electric field is applied to a bulk semiconductor, a further redshift in the absorption spectrum is observed due to Franz–Keldysh effect . Due to their opposite electric charges, the electron and the hole constituting the exciton will be pulled apart under the influence of the external electric field. If the field is strong enough
then excitons cease to exist in the bulk material. This somewhat limits the applicability of Franz-Keldysh for modulation purposes, as the redshift induced by the applied electric field is countered by shift towards higher energies due to the absence of exciton generations.
This problem does not exist in QCSE, as electrons and holes are confined in the quantum wells. As long as the quantum well depth is comparable to the excitonic Bohr radius , strong excitonic effects will be present no matter the magnitude of the applied electric field. Furthermore, quantum wells behave as two dimensional systems, which strongly enhance excitonic effects with respect to bulk material. In fact, solving the Schrödinger equation for a Coulomb potential in a two dimensional system yields an excitonic binding energy of
which is four times as high as the three dimensional case for the 1 s {\displaystyle 1s} solution. [ 6 ]
Quantum-confined Stark effect's most promising application lies in its ability to perform optical modulation in the near infrared spectral range, which is of great interest for silicon photonics and down-scaling of optical interconnects . [ 2 ] [ 7 ] A QCSE based electro-absorption modulator consists of a PIN structure where the instrinsic region contains multiple quantum wells and acts as a waveguide for the carrier signal . An electric field can be induced perpendicularly to the quantum wells by applying an external, reverse bias to the PIN diode, causing QCSE. This mechanism can be employed to modulate wavelengths below the band gap of the unbiased system and within the reach of the QCSE induced redshift.
Although first demonstrated in GaAs / Al x Ga 1-x As quantum wells, [ 1 ] QCSE started to generate interest after its demonstration in Ge / SiGe . [ 8 ] Differently from III/V semiconductors, Ge/SiGe quantum well stacks can be epitaxially grown on top of a silicon substrate, provided the presence of some buffer layer in between the two. This is a decisive advantage as it allows Ge/SiGe QCSE to be integrated with CMOS technology [ 9 ] and silicon photonics systems.
Germanium is an indirect gap semiconductor, with a bandgap of 0.66 eV . However it also has a relative minimum in the conduction band at the Γ {\displaystyle \Gamma } point , with a direct bandgap of 0.8 eV, which corresponds to a wavelength of 1550 nm . QCSE in Ge/SiGe quantum wells can therefore be used to modulate light at 1.55 μ m {\displaystyle \mu m} , [ 9 ] which is crucial for silicon photonics applications as 1.55 μ m {\displaystyle \mu m} is the optical fiber `s transparency window and the most extensively employed wavelength for telecommunications.
By fine tuning material parameters such as quantum well depth, biaxial strain and silicon content in the well, it is also possible to tailor the optical band gap of the Ge/SiGe quantum well system to modulate at 1310 nm, [ 9 ] [ 10 ] which also corresponds to a transparency window for optical fibers.
Electro-optic modulation by QCSE using Ge/SiGe quantum wells has been demonstrated up to 23 GHz with energies per bit as low as 108 fJ. [ 11 ] and integrated in a waveguide configuration on a SiGe waveguide [ 12 ] | https://en.wikipedia.org/wiki/Quantum-confined_Stark_effect |
Quantum 1/f noise is an intrinsic and fundamental part of quantum mechanics . Fighter pilots, photographers, and scientists all appreciate the higher quality of images and signals resulting from the consideration of quantum 1/f noise. Engineers have battled unwanted 1/f noise since 1925, giving it poetic names (such as flicker noise, funkelrauschen, bruit de scintillation, etc.) due to its mysterious nature. The Quantum 1/f noise theory was developed about 50 years later, describing the nature of 1/f noise, allowing it to be explained and calculated via straightforward engineering formulas. It allows for the low-noise optimization of materials, devices and systems of most high-technology applications of modern industry and science. The theory includes the conventional and coherent quantum 1/f effects (Q1/fE). Both effects are combined in a general engineering formula, and present in Q1/f noise, which is itself most of fundamental 1/f noise. The latter is defined as the result of the simultaneous presence of nonlinearity and a certain type of homogeneity in a system, and can be quantum or classical.
The conventional Q1/fE represents 1/f fluctuations caused by bremsstrahlung , decoherence and interference in the scattering of charged particles off one another, in tunneling or in any other process in solid state physics and in general.
It has also recently been claimed that 1/f noise has been seen in higher ordered self constructing functions, as well as complex systems, both biological, chemical, and physical. [ citation needed ]
The basic derivation of quantum 1/f was made by Peter Handel, a theoretical physicist at the University of Missouri–St. Louis , and published in Physical Review A , in August 1980.
Several hundred papers [ vague ] have been published by many authors [ vague ] on Handel's quantum theory on 1/f noise, which is a new aspect of quantum mechanics. They verified, applied, and further developed the quantum 1/f noise formulas. [ 1 ] Aldert van der Ziel , the nestor of the electronic noise field, verified and applied it in many devices and systems, together with dozens of his PhD students. It is described in the last of his 12 books: "Noise in electronic devices and circuits" published by Wiley in 1986. He also updated and generalized many verifications, practical applications, etc., in his authoritative 1988 review "Unified Description of 1/f Noise" in Proceedings of IEEE . [ 2 ]
In 1986 and 1987, two independent groups of theorists of the field, Group-1: Theo Nieuwenhuizen, Daan Frenkel and Nico G. van Kampen ; Group-2: Laszlo B. Kish and Peter Heszler ; concluded that Handel's theory explaining the quantum 1/f effect was incorrect for both physical and mathematical reasons. [ 3 ] [ 4 ] Shortly thereafter an independent set of arguments showing that the "quantum 1/f noise" explanation of electronic 1/f noise was certainly incorrect was included in a standard review article on 1/f noise by Michael Weissman. [ 5 ] Nieuwenhuizen, et al., state in the conclusion of their paper, "As the theoretical basis for Handel's quantum theory of 1/f noise appears to be lacking, we must conclude that the agreement with experiments is fortuituous" [ 3 ] and, in this way, they are indicating that some of the published experimental results are suspicious. Though there have been attempts to answer some of the objections to Handel's theory, quantum 1/f noise is considered to be a non-existent effect by the majority of scientists that are familiar with its theory. [ citation needed ] The difficulty is that here a judgment based on fundamental science requires the knowledge of quantum electrodynamics however most of noise scientists are solid state physicists or engineers. Science citation index shows over 20 thousand papers annually with "noise" and/or "fluctuation"(s) keywords. The opinion of the above-mentioned relevant experts in the field of noise is that, until the publication rate on the non-existent quantum 1/f noise effect stays around 1 paper/year, it is more economical to refer to the old denials [ 3 ] [ 4 ] than to write up new refusals.
For more on Quantum 1/f noise, see:
For the coherent quantum 1/f effect, see: | https://en.wikipedia.org/wiki/Quantum_1/f_noise |
Quantum Aspects of Life , a book published in 2008 with a foreword by Roger Penrose , explores the open question of the role of quantum mechanics at molecular scales of relevance to biology. The book contains chapters written by various world-experts from a 2003 symposium and includes two debates from 2003 to 2004; giving rise to a mix of both sceptical and sympathetic viewpoints. The book addresses questions of quantum physics , biophysics , nanoscience , quantum chemistry , mathematical biology , complexity theory , and philosophy that are inspired by the 1944 seminal book What Is Life? by Erwin Schrödinger .
Section 1: Emergence and Complexity
Section 2: Quantum Mechanisms in Biology
Section 3: The Biological Evidence
Section 4: Artificial Quantum Life
Section 5: The Debate | https://en.wikipedia.org/wiki/Quantum_Aspects_of_Life |
The quantum Boltzmann equation, also known as the Uehling–Uhlenbeck equation , [ 1 ] [ 2 ] is the quantum mechanical modification of the Boltzmann equation , which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by L.W. Nordheim (1928), [ 3 ] and by and E. A. Uehling and George Uhlenbeck (1933). [ 4 ]
In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation. [ ∂ ∂ t + v ⋅ ∇ x + F ⋅ ∇ p ] f ( x , p , t ) = Q [ f ] ( x , p ) {\displaystyle \left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla _{x}+\mathbf {F} \cdot \nabla _{p}\right]f(\mathbf {x} ,\mathbf {p} ,t)={\mathcal {Q}}[f](\mathbf {x} ,\mathbf {p} )}
where F {\displaystyle \mathbf {F} } represents an externally applied potential acting on the gas' p-space distribution and Q {\displaystyle {\mathcal {Q}}} is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of Q {\displaystyle {\mathcal {Q}}} , which depends on the physics of the system to be modeled. [ 5 ]
The quantum Boltzmann equation gives irreversible behavior, and therefore an arrow of time ; that is, after a long enough time it gives an equilibrium distribution which no longer changes. Although quantum mechanics is microscopically time-reversible, the quantum Boltzmann equation gives irreversible behavior because phase information is discarded [ 6 ] only the average occupation number of the quantum states is kept. The solution of the quantum Boltzmann equation is therefore a good approximation to the exact behavior of the system on time scales short compared to the Poincaré recurrence time , which is usually not a severe limitation, because the Poincaré recurrence time can be many times the age of the universe even in small systems.
The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in semiconductor optics. [ 7 ] For example, the energy distribution of a gas of excitons as a function of time (in picoseconds), measured using a streak camera, has been shown [ 8 ] to approach an equilibrium Maxwell-Boltzmann distribution .
A typical model of a semiconductor may be built on the assumptions that:
Considering the exchange of momentum q {\displaystyle \mathbf {q} } between electrons with initial momenta k {\displaystyle \mathbf {k} } and k 1 {\displaystyle \mathbf {k_{1}} } , it is possible to derive the expression Q [ f ] ( k ) = − 2 ℏ ( 2 π ) 5 ∫ d q ∫ d k 1 | v ^ ( q ) | 2 δ ( ℏ 2 2 m ( | k − q | 2 + | k 1 + q | 2 − k 1 2 − k 2 ) ) [ f k f k 1 ( 1 − f k − q ) ( 1 − f k 1 + q ) − f k − q f k 1 + q ( 1 − f k ) ( 1 − f k 1 ) ] {\displaystyle {\mathcal {Q}}[f](\mathbf {k} )={\frac {-2}{\hbar (2\pi )^{5}}}\int d\mathbf {q} \int d\mathbf {k_{1}} |{\hat {v}}(\mathbf {q} )|^{2}\delta \left({\frac {\hbar ^{2}}{2m}}(|\mathbf {k-q} |^{2}+|\mathbf {k_{1}+q} |^{2}-\mathbf {k} _{1}^{2}-\mathbf {k} ^{2})\right)\left[f_{\mathbf {k} }f_{\mathbf {k_{1}} }(1-f_{\mathbf {k-q} })(1-f_{\mathbf {k_{1}+q} })-f_{\mathbf {k-q} }f_{\mathbf {k_{1}+q} }(1-f_{\mathbf {k} })(1-f_{\mathbf {k_{1}} })\right]} | https://en.wikipedia.org/wiki/Quantum_Boltzmann_equation |
The Quantum Chemistry Program Exchange (QCPE) was an organization located at Indiana University Bloomington from 1963 to 2007 that was devoted to the distribution of computational chemistry software before electronic file transfer on the internet became a widely available method of software distribution. [ 1 ] The QCPE was originally founded by Prof. Harrison Shull [ 2 ] and was managed by Richard Counts for most of its existence. Financial support for the QCPE was originally provided by the Air Force Office of Scientific Research until 1969, and funding continued under an interim grant from the National Science Foundation in 1971 until it became financially self-sustaining in 1973. [ 1 ]
The QCPE maintained a catalog of software that expanded through regular contributions from chemistry software developers. [ 3 ] New software contributions were announced through a quarterly QCPE Newsletter [ 4 ] that were eventually formalized into a QCPE Bulletin [ 5 ] in 1981, which allowed for software citations to numbered software entries in the Bulletin that announced their release. QCPE members paid for subscriptions to the Newsletter/Bulletin and additionally paid a processing and delivery fee to receive software from the QCPE catalog. The software distribution options expanded alongside technological development, starting from punched cards and magnetic tape drives delivered by mail , before adopting floppy disks and CD-ROMs , and eventually electronic delivery by FTP . The QCPE grew rapidly in its early days, with about 400 members and a catalog of nearly 100 programs after its first 3 years of operation. [ 6 ] In the 1980's and early 1990's, the QCPE also organized annual summer workshops to train scientists in the use of its more popular software. At its peak in the mid-1980's, the QCPE had over 2000 members, over 400 programs available, and an annual income near $400,000. [ 1 ]
The most visible legacy of the QCPE are the thousands of software citations to the QCPE Bulletin in scientific publications over 4 decades, with a peak of over 1000 per year in the early 1990's. [ 1 ] The most popular software in the early days of the QCPE was GAUSSIAN (QCPE #236, #368, #406) [ 7 ] before it was removed from the QCPE catalog to become commercial software, and the most popular software in its later years was MOPAC (QCPE #455, #688, #689). [ 1 ] Other popular software distributed by the QCPE included POLYATOM (QCPE #47, #199), CNDO/2 (QCPE #91), AMPAC (QCPE #506), CRYSTAL (QCPE #577), Molden (QCPE #619), and MM2 / MM3 (QCPE #690-#698). [ 8 ] | https://en.wikipedia.org/wiki/Quantum_Chemistry_Program_Exchange |
In quantum mechanics , the quantum Cheshire cat is a quantum phenomena that suggests that a particle's physical properties can take a different trajectory from that of the particle itself. The name makes reference to the Cheshire Cat from Lewis Carroll 's Alice's Adventures in Wonderland , a feline character which could disappear leaving only its grin behind. The effect was originally proposed by Yakir Aharonov , Daniel Rohrlich, Sandu Popescu and Paul Skrzypczyk in 2012. [ 1 ]
In classical physics , physical properties cannot be detached from the object associated to it. If a magnet follows a given trajectory in space and time, its magnetic moment follows it through the same trajectory. However, in quantum mechanics, particles can be in a quantum superposition of more than one trajectory previous to measurement . The quantum Cheshire experiments suggests that previous to a measurement , a particle may take two paths, but the property of the particle, like the spin of a massive particle or the polarization of a light beam, travels only through one of the paths, while the particle takes the opposite path. The conclusion is only obtained from an analysis of weak measurements , which consist in interpreting the particle history previous to measurement by studying quantum systems in the presence of small disturbances.
Experimental demonstration of the quantum Cheshire cat have already been claimed in different systems, including photons [ 2 ] and neutrons . [ 3 ] The effect has been suggested as a probe to study properties of massive particles by detaching it from its magnetic moment in order to shield them from electromagnetic disturbances. [ 4 ] [ 5 ] A dynamical quantum Cheshire cat has also been proposed as a counterfactual quantum communication protocol. [ 6 ]
Neutrons are uncharged subatomic particles that have a magnetic moment , with two possible projections on any given axis.
A beam of neutrons, with all with their magnetic moments aligned to the right, enters a Mach–Zehnder interferometer coming from the left-to-right. The neutrons can exit the interferometer into a right port, where a detector of neutrons with right magnetic moment is located, or upwards into a dark port with no detector (see picture). [ 7 ]
The neutrons enter the interferometer and reach a beam splitter . Each neutron that passes through, enters into a quantum superposition state of two different paths, namely A and B . This initial state is referred to as the preselected state. As the neutrons travel the different paths, their wave functions reunites at a second beam splitter, causing interference. If there is nothing in the path of the neutrons, every neutron exits to the interferometer moving to the right and activates the detector. [ 7 ] No neutron escapes upwards into the dark port due to destructive interference.
One can add different components and filters in one of the paths. By adding a filter that flips the magnetic moment of the neutron in path B (lower branch), it leads to a new superposition state: neutron taking path A with a magnetic moment pointing right, plus the neutron taking path B with the magnetic moment flipped pointing to the left. This state is called a postselected state. [ 7 ] As the states cannot longer interfere coherently due to this modification, the neutrons can exit through the two ports, either to the right reaching the detector or exiting towards the dark port.
In this configuration, if the detector clicks, it is only because the neutrons had a magnetic moment oriented in to the right. By means of this postselection , it can be confidently stated that the neutron that reached the detector passed through path A , which is the only path to contains neutron magnetic moments oriented to the right. This effect can be easily demonstrated by putting a thin absorber of neutrons in the path. [ 7 ] By placing the absorber in path B , the rate of neutrons that are detected remains constant. However, when the absorber is positioned in path A, the detection rate decreases, providing evidence that detected neutrons in the postselected state travel only through path A . [ 7 ]
If a magnetic field is applied perpendicular to the plane of the interferometer and localized in either path A or path B , the number of neutrons that are detected changes, as the magnetic fields makes the neutrons precess and alters the probabilities of being measured. Additionally, measuring the magnetism and the trajectory (with an absorber) at the same time is not possible without also disrupting the quantum state.
The quantum Cheshire cat appears in the weak limit of the interaction. When a sufficiently small magnetic field is applied to path A , there is no impact on the measurement. In contrast, if the magnetic field is applied to path B , the detection rate diminishes, demonstrating that the neutrons magnetism, perpendicular to the plane of the interferometer, predominantly resided in path B . [ 7 ] We can do the same with a thin absorber, showing that only the neutrons that are detected are all from path A . This experiment effectively separated the "cat", representing the neutron, from its "grin", symbolizing its magnetic moment out of the plane. [ 7 ]
Consider a particle with a two-level property that can be either | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } , this can be for example the horizontal and vertical polarization of a photon or the spin projection of a spin-1/2 particle as in the previous example with the neutrons. One of these two polarization states (let's say | 0 ⟩ {\displaystyle |0\rangle } ) is chosen and the particle is then prepared to be in the following superposition: [ 1 ]
where | A ⟩ {\displaystyle |A\rangle } and | B ⟩ {\displaystyle |B\rangle } are two possible orthogonal trajectories of the particle. The state | Ψ ⟩ {\displaystyle |\Psi \rangle } is called the preselected state.
A filter is added in path | B ⟩ {\displaystyle |B\rangle } of the particle in order to flip its polarization from | 0 ⟩ {\displaystyle |0\rangle } to | 1 ⟩ {\displaystyle |1\rangle } , such that it ends up in the state [ 1 ]
such state indicates that if the particle is measured to be in state | 0 ⟩ {\displaystyle |0\rangle } , the particle took path | A ⟩ {\displaystyle |A\rangle } ; analogously, if the particle is measured to be in state | 1 ⟩ {\displaystyle |1\rangle } , the particle took path | B ⟩ {\displaystyle |B\rangle } . The state | Φ ⟩ {\displaystyle |\Phi \rangle } is called the postselected state.
Using postselection techniques, the particle is measured in order to detect the overlap between the preselected state and postselected state. If there are no disturbances, the preselected and postselected states produce the same results 1/4 of time.
We define the weak value of an operator O {\displaystyle O} given by [ 8 ]
where | ψ ⟩ {\displaystyle |\psi \rangle } is the preselected state and | ϕ ⟩ {\displaystyle |\phi \rangle } the postselected state. This calculation can be thought as the contribution of a given interaction up to linear order.
For the system, one considers two projectors operators given by
and
which measure if the particle is on either path | A ⟩ {\displaystyle |A\rangle } or | B ⟩ {\displaystyle |B\rangle } , respectively.
Additionally, an out-of the-plane polarization operator is defined as
this operator can be thought as a measure of angular momentum in the system. [ 1 ] Outside the weak limit, the interaction related to this operator tends to make the polarization precess between | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } .
Performing the following weak measurements on the positions with | ϕ ⟩ = | Φ ⟩ {\displaystyle |\phi \rangle =|\Phi \rangle } and | ψ ⟩ = | Ψ ⟩ {\displaystyle |\psi \rangle =|\Psi \rangle } , one obtains the following
These weak values indicate that if the path | A ⟩ {\displaystyle |A\rangle } is slightly perturbed, then the measurement is perturbed. While if instead path | B ⟩ {\displaystyle |B\rangle } is perturbed this does not affect the measurement.
We also consider weak measurements on the out-of the-plane polarization in each of the paths, such that
These values indicate that if the polarization is slightly modified in path | B ⟩ {\displaystyle |B\rangle } , then the results are slightly modified too. However, if the polarization is perturbed in path | A ⟩ {\displaystyle |A\rangle } there is no correction to the intensity measured (in the weak limit).
These 4 weak values lead to the quantum Cheshire cat conclusion.
The proposal of quantum Cheshire cat has received some criticism. [ 9 ] Popescu, one of the authors of the original paper, acknowledged it was not well received by all of the referees who first reviewed the original work. [ 9 ]
As the quantum Cheshire cat effect is subjected to analysis of the trajectory before measurement, its conclusion depends on the interpretation of quantum mechanics , which is still an open problem in physics. Some authors reach different conclusions for this effect or disregard the effect completely. [ 10 ] It has been suggested that the quantum Cheshire cat is just an apparent paradox raising from misinterpreting wave interference. [ 11 ] Other authors consider that it can be reproduced classically. [ 12 ] [ 10 ]
The experimental results depend on the postselection and analysis of the data. It has been suggested that the weak value cannot be interpreted as a real property of the system, but as an optimal estimate of the corresponding observable, given that the postselection is successful. [ 3 ] Aephraim M. Steinberg , notes that the experiment with neutrons does not prove that any single neutron took a different path than its magnetic moments; but shows only that the measured neutrons behaved this way on average. [ 13 ] It has also been argued that even if the weak values were measured in the neutron Cheshire cat experiment, they do not imply that a particle and one of its properties have been disembodied due to unavoidable quadratic interactions in the experiment. [ 14 ] [ 15 ] [ 10 ] This last point was acknowledged by A. Matzkin, one of the coauthors of the neutron experiment paper. [ 15 ] | https://en.wikipedia.org/wiki/Quantum_Cheshire_cat |
Quantum Computation and Quantum Information is a textbook about quantum information science written by Michael Nielsen and Isaac Chuang , regarded as a standard text on the subject. [ 1 ] It is informally known as " Mike and Ike ", after the candies of that name . [ 2 ] The book assumes minimal prior experience with quantum mechanics and with computer science, aiming instead to be a self-contained introduction to the relevant features of both. ( Lov Grover recalls a postdoc disparaging it with the remark, "The book is too elementary – it starts off with the assumption that the reader does not even know quantum mechanics." [ 3 ] ) The focus of the text is on theory, rather than the experimental implementations of quantum computers, which are discussed more briefly. [ 4 ]
As of December 2024 [update] , the book has been cited over 58,000 times on Google Scholar . [ 5 ] In 2019, Nielsen adapted parts of the book for his Quantum Country project. [ 6 ]
Peter Shor called the text "an excellent book". Lov Grover called it "the bible of the quantum information field". Scott Aaronson said about it, " 'Mike and Ike' as it's affectionately called, remains the quantum computing textbook to which all others are compared." [ 7 ] David DiVincenzo said, "More than any of the previous attempts, this book has identified the essential foundations of quantum information theory with a clarity that has even, in a few cases, permitted the authors to obtain some original results and point toward new research directions." [ 8 ] A review in the November 2001 edition of Foundations of Physics says, "Among the handful of books that have been written on this new subject, the present volume is the most complete and comprehensive." [ 9 ] | https://en.wikipedia.org/wiki/Quantum_Computation_and_Quantum_Information |
Quantum Computing: A Gentle Introduction is a textbook on quantum computing . It was written by Eleanor Rieffel and Wolfgang Polak, and published in 2011 by the MIT Press .
Although the book approaches quantum computing through the model of quantum circuits , [ 1 ] [ 2 ] it is focused more on quantum algorithms than on the construction of quantum computers. [ 2 ] It has 13 chapters, divided into three parts: "Quantum building blocks" (chapters 1–6), "Quantum algorithms" (chapters 7–9), and "Entangled subsystems and robust quantum computation" (chapters 10–13). [ 3 ]
After an introductory chapter overviewing related topics including quantum cryptography , quantum information theory , and quantum game theory , chapter 2 introduces quantum mechanics and quantum superposition using polarized light as an example, also discussing qubits , the Bloch sphere representation of the state of a qubit, and quantum key distribution . Chapter 3 introduces direct sums , tensor products , and quantum entanglement , and chapter 4 includes the EPR paradox , Bell's theorem on the impossibility of local hidden variable theories, as quantified by Bell's inequality. Chapter 5 discusses unitary operators , quantum logic gates , quantum circuits , and functional completeness for systems of quantum gates. Chapter 6, the final chapter of the building block section, discusses (classical) reversible computing , and the conversion of arbitrary computations to reversible computations, a necessary step to performing them on quantum devices. [ 2 ] [ 3 ]
In the section of the book on quantum algorithms, chapter 7 includes material on quantum complexity theory and the Deutch algorithm, Deutsch–Jozsa algorithm , Bernstein–Vazirani algorithm , and Simon's algorithm , algorithms devised to prove separations in quantum complexity by solving certain artificial problems faster than could be done classically. It also covers the quantum Fourier transform . Chapter 8 covers Shor's algorithm for integer factorization , and introduces the hidden subgroup problem . Chapter 9 covers Grover's algorithm and the quantum counting algorithm for speeding up certain kinds of brute-force search . The remaining chapters return to the topic of quantum entanglement and discuss quantum decoherence , quantum error correction , and its use in designing robust quantum computing devices, with the final chapter providing an overview of the subject and connections to additional topics. Appendices provide a graphical approach to tensor products of probability spaces, and extend Shor's algorithm to the abelian hidden subgroup problem. [ 2 ] [ 3 ]
The book is suitable as an introduction to quantum computing for computer scientists, mathematicians, and physicists, requiring of them only a background in linear algebra and the theory of complex numbers , [ 2 ] [ 3 ] although reviewer Donald L. Vestal suggests that additional background in the theory of computation , abstract algebra , and information theory would also be helpful. [ 4 ] Prior knowledge of quantum mechanics is not required. [ 2 ]
Reviewer Kyriakos N. Sgarbas has some minor notational quibbles with the book's presentation, and complains that the level of difficulty is uneven and that it lacks example solutions. [ 2 ] However, reviewer Valerio Scarani calls the book "a masterpiece", particularly praising it for its orderly arrangement, its well-thought-out exercises, the self-contained nature of its chapters, and its inclusion of material warning readers against falling into common pitfalls. [ 1 ]
There are many other textbooks on quantum computing; [ 2 ] for instance, Scarani lists Quantum Computer Science: An Introduction by N. David Mermin (2007), An Introduction to Quantum Computing by Kaye, Laflamme, and Mosca (2007), and A Short Introduction to Quantum Information and Quantum Computation by Michel Le Bellac (2006). [ 1 ] Sgarbas lists in addition Quantum Computing Explained by D. McMahon (2008) and Quantum Computation and Quantum Information by M. A. Nielsen and I. L. Chuang (2000). [ 2 ] | https://en.wikipedia.org/wiki/Quantum_Computing:_A_Gentle_Introduction |
Quantum Computing Since Democritus is a 2013 book on quantum information science written by Scott Aaronson . [ 1 ] It is loosely based on a course Aaronson taught at the University of Waterloo , Canada, the lecture notes for which are available online. [ 2 ]
Aaronson has stated that he intends the book to be at the same level as Leonard Susskind 's The Theoretical Minimum or Roger Penrose 's The Road to Reality ; [ 3 ] Physics Today compared it to George Gamow 's One Two Three... Infinity . [ 4 ] The book covers everything from computer science to mathematics to quantum mechanics and quantum computing , starting, as the title indicates, with Democritus .
The front cover image is an oil canvas painting of Democritus by Hendrik ter Brugghen dated 1628. [ 5 ] It depicts Democritus as a young, laughing hedonist who points in the distance, as to where the folly of mankind is found.
The image invokes Aaronson's discussions [ 6 ] on Democritus' concept of atoms and the void, which forms the foundational understanding of matter at the atomic level, is relevant to quantum computing, where manipulating and controlling individual quantum objects for calculations echoes the early atomic theory's significance.
Scott Aaronson is a professor of theoretical computer science at the University of Texas at Austin . He was previously a member of the faculty at MIT . [ 7 ]
In the Journal of the American Mathematical Society , Avi Wigderson considered it to have "much insight, wisdom, and fun", but conceded that it "is not for everyone". Widgerson noted in particular that the book would have been easier to read if it had provided more background material, and that it had little in the way of references to prior literature. [ 8 ] Reviewing the book for Physics Today , Francis Sullivan deemed it "stimulating", while saying that it "covers too much territory to be used as a textbook" and taking exception with Aaronson's attitude "that mathematicians like complication because it makes things more interesting". [ 4 ] Frederic Green's enthusiastic review for SIGACT News also judged the book poorly suited for a classroom text, except possibly in "a seminar-style course with a fairly open structure". [ 9 ]
Reviel Netz gave the book a positive review in Common Knowledge , quipping that "I suspect that I was sent this book by mistake; despite its title, it has nothing to do with ancient science, my field." [ 10 ] | https://en.wikipedia.org/wiki/Quantum_Computing_Since_Democritus |
Quantum ESPRESSO ( Quantum Open-Source Package for Research in Electronic Structure, Simulation, and Optimization ; QE ) [ 2 ] [ 3 ] is a suite for first-principles electronic-structure calculations and materials modeling, distributed for free and as free software under the GNU General Public License . It is based on density functional theory (DFT), plane wave basis sets , and pseudopotentials (both norm-conserving and ultrasoft).
The core plane wave DFT functions of QE are provided by the PWscf component (PWscf previously existed as an independent project). PWscf ( Plane-Wave Self-Consistent Field ) is a set of programs for electronic structure calculations within DFT and density functional perturbation theory, using plane wave basis sets and pseudopotentials . The software is released under the GNU General Public License .
The latest stable version QE-7.4.1 was released on 14 March 2025.
Quantum ESPRESSO is an open initiative of the CNR-IOM DEMOCRITOS National Simulation Center in Trieste ( Italy ) and its partners, in collaboration with different centers worldwide such as MIT , Princeton University , the University of Minnesota and the École Polytechnique Fédérale de Lausanne . The project is coordinated by the QUANTUM ESPRESSO foundation, which was formed by many research centers and groups all over the world. The first version, called pw.1.0.0 , was released on 15-06-2001.
The program is written mainly in Fortran-90 with some parts in C or in Fortran-77. It is composed of a set of core components, a set of plug-ins for advanced tasks, and a set of third-party packages.
The basic packages include Pwscf , [ 4 ] which solves the self-consistent Kohn-Sham equations , obtained for a periodic solid, CP to carry out Car-Parrinello molecular dynamics , and PostProc , which allows data analysis and plotting. Noteworthy additional packages include atomic for pseudopotential generation, PHonon for density-functional perturbation theory (DFPT) and the calculation of second- and third-order derivatives of the energy with respect to atomic displacements, and NEB (nudged elastic band) for the calculation of reaction pathways and energy barriers.
The different tasks that can be performed include
The main components of the Quantum ESPRESSO distribution are designed to exploit the architecture of today's supercomputers, which are characterized by multiple levels and layers of inter-processor communication. Parallelization is achieved using both MPI and OpenMP , allowing the main codes of the distribution to run in parallel on most or all parallel machines with very good performance. In recent years of development, Quantum ESPRESSO has increasingly adopted CUDA -basec GPU acceleration across the different tools to improve performance. | https://en.wikipedia.org/wiki/Quantum_ESPRESSO |
Quantum Experiment using Satellite Technology was launched in 2017 by the Raman Research Institute . In February 2021, the project demonstrated quantum communication for 50 m apart, and on 19 March 2021 for 300 m apart inline of sight in Space Applications Centre , which was done in coordination with the Indian Space Research Organisation , Indian Institute of Science and Tata Institute of Fundamental Research . [ 1 ] [ 2 ] [ 3 ] [ 4 ] Quantum Experiment using Satellite Technology is India's first project on satellite based long distance quantum communication. [ 5 ] [ 6 ] [ 7 ] [ 8 ]
This spacecraft or satellite related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_Experiments_using_Satellite_Technology |
The Quantum Technologies Flagship is a European Union scientific research initiative. [ 1 ] With a budget of €1 billion, it is one of the large scale initiatives organized by the Future and Emerging Technologies program, along with the Human Brain Project and the Graphene Flagship .The Quantum Flagship funds over 5,000 Europeans researchers over ten years. Its long-term vision is to develop in Europe a quantum web, where quantum computers, simulators and sensors are interconnected via quantum communication networks. The objective being to develop in Europe a competitive quantum industry making research results available as commercial applications and disruptive technologies. [ 2 ]
In 2016, the European Commissioner for digital economy and society Günther Oettinger invited the quantum community to provide a common European strategy on quantum technologies . [ 3 ] Endorsed by 3.400 individuals from academia and industry, a “quantum manifesto” [ 4 ] was released in May 2016 in the Netherlands [ 5 ] The document called for the European Commission to launch an ambitious European initiative in quantum technologies to ensure Europe's leading role in a technological revolution underway.
Following this document, the European Commission appointed a High Level Steering Committee, composed of 12 academic members and 12 industry members, to deliver a strategic research agenda, an implementation model and a governance model for the quantum technologies flagship. The report of this committee was delivered in 2017, [ 6 ] giving a long-term vision of a “quantum web” and the flagship was launched in 2018, bringing under the same brand research institutions, industry, and public funders. The quantum technologies flagship with an expected budget of €1 billion from the EU over 10 years aims to support the transformation of European research into commercial applications. [ 1 ]
In October 2018, the recipients of the first grants were announced, for a total of €132 million distributed among 20 projects and three years. [ 7 ] | https://en.wikipedia.org/wiki/Quantum_Flagship |
The quantum Hall effect (or integer quantum Hall effect ) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields , in which the Hall resistance R xy exhibits steps that take on the quantized values
where V Hall is the Hall voltage , I channel is the channel current , e is the elementary charge and h is the Planck constant . The divisor ν can take on either integer ( ν = 1, 2, 3,... ) or fractional ( ν = 1 / 3 , 2 / 5 , 3 / 7 , 2 / 3 , 3 / 5 , 1 / 5 , 2 / 9 , 3 / 13 , 5 / 2 , 12 / 5 ,... ) values. Here, ν is roughly but not exactly equal to the filling factor of Landau levels . The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether ν is an integer or fraction, respectively.
The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the electron density remains constant when the Fermi level is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized (see Anderson localization ). [ 1 ]
The fractional quantum Hall effect is more complicated and still considered an open research problem. [ 2 ] Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without Landau levels . [ 3 ] This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the quantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents. [ 4 ]
The quantization of the Hall conductance ( G x y = 1 / R x y {\displaystyle G_{xy}=1/R_{xy}} ) has the important property of being exceedingly precise. [ 5 ] Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e 2 / h to better than one part in a billion. [ 6 ] It has allowed for the definition of a new practical standard for electrical resistance , based on the resistance quantum given by the von Klitzing constant R K . This is named after Klaus von Klitzing , the discoverer of exact quantization. The quantum Hall effect also provides an extremely precise independent determination of the fine-structure constant , a quantity of fundamental importance in quantum electrodynamics .
In 1990, a fixed conventional value R K-90 = 25 812 .807 Ω was defined for use in resistance calibrations worldwide. [ 7 ] Later, the 2019 revision of the SI fixed exact values of h and e , resulting in an exact R K = h / e 2 = 25 812 .807 45 ... Ω . [ 8 ]
The fractional quantum Hall effect is considered part of exact quantization . [ 9 ] Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle of gauge invariance together with another symmetry (see Anomalies ). The integer quantum Hall effect instead is considered a solved research problem [ 10 ] [ 11 ] and understood in the scope of TKNN formula and Chern–Simons Lagrangians .
The fractional quantum Hall effect is still considered an open research problem. [ 2 ] The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of charge–flux composites known as composite fermions . [ 12 ] Other models to explain the fractional quantum Hall effect also exists. [ 13 ] Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope of composite fermions and Non Abelian Chern–Simons Lagrangians .
In 1957, Carl Frosch and Lincoln Derick were able to manufacture the first silicon dioxide field effect transistors at Bell Labs , the first transistors in which drain and source were adjacent at the surface. [ 14 ] Subsequently, a team demonstrated a working MOSFET at Bell Labs 1960. [ 15 ] [ 16 ] This enabled physicists to study electron behavior in a nearly ideal two-dimensional gas . [ 17 ]
In a MOSFET, conduction electrons travel in a thin surface layer, and a " gate " voltage controls the number of charge carriers in this layer. This allows researchers to explore quantum effects by operating high-purity MOSFETs at liquid helium temperatures. [ 17 ]
The integer quantization of the Hall conductance was originally predicted by University of Tokyo researchers Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. [ 18 ] In 1978, the Gakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. [ 19 ]
In 1980, Klaus von Klitzing , working at the high magnetic field laboratory in Grenoble with silicon -based MOSFET samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall resistance was exactly quantized. [ 20 ] [ 17 ] For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics . A link between exact quantization and gauge invariance was subsequently proposed by Robert Laughlin , who connected the quantized conductivity to the quantized charge transport in a Thouless charge pump. [ 11 ] [ 21 ] Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures , although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in graphene at temperatures as high as room temperature, [ 22 ] and in the magnesium zinc oxide ZnO–Mg x Zn 1− x O. [ 23 ]
In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. To determine the values of the energy levels the Schrödinger equation must be solved.
Since the system is subjected to a magnetic field, it has to be introduced as an electromagnetic vector potential in the Schrödinger equation . The system considered is an electron gas that is free to move in the x and y directions, but is tightly confined in the z direction. Then, a magnetic field is applied in the z direction and according to the Landau gauge the electromagnetic vector potential is A = ( 0 , B x , 0 ) {\displaystyle \mathbf {A} =(0,Bx,0)} and the scalar potential is ϕ = 0 {\displaystyle \phi =0} . Thus the Schrödinger equation for a particle of charge q {\displaystyle q} and effective mass m ∗ {\displaystyle m^{*}} in this system is:
where p {\displaystyle \mathbf {p} } is the canonical momentum, which is replaced by the operator − i ℏ ∇ {\displaystyle -i\hbar \nabla } and ε {\displaystyle \varepsilon } is the total energy.
To solve this equation it is possible to separate it into two equations since the magnetic field just affects the movement along x and y axes. The total energy becomes then, the sum of two contributions ε = ε z + ε x y {\displaystyle \varepsilon =\varepsilon _{z}+\varepsilon _{xy}} . The corresponding equations in z axis is:
To simplify things, the solution V ( z ) {\displaystyle V(z)} is considered as an infinite well. Thus the solutions for the z direction are the energies ε z = n z 2 π 2 ℏ 2 2 m ∗ L 2 {\textstyle \varepsilon _{z}={\frac {n_{z}^{2}\pi ^{2}\hbar ^{2}}{2m^{*}L^{2}}}} , n z = 1 , 2 , 3... {\displaystyle n_{z}=1,2,3...} and the wavefunctions are sinusoidal. For the x {\displaystyle x} and y {\displaystyle y} directions, the solution of the Schrödinger equation can be chosen to be the product of a plane wave in y {\displaystyle y} -direction with some unknown function of x {\displaystyle x} , i.e., ψ x y = u ( x ) e i k y y {\displaystyle \psi _{xy}=u(x)e^{ik_{y}y}} . This is because the vector potential does not depend on y {\displaystyle y} and the momentum operator p ^ y {\displaystyle {\hat {p}}_{y}} therefore commutes with the Hamiltonian. By substituting this Ansatz into the Schrödinger equation one gets the one-dimensional harmonic oscillator equation centered at x k y = ℏ k y e B {\textstyle x_{k_{y}}={\frac {\hbar k_{y}}{eB}}} .
where ω c = e B m ∗ {\textstyle \omega _{\rm {c}}={\frac {eB}{m^{*}}}} is defined as the cyclotron frequency and l B 2 = ℏ e B {\textstyle l_{B}^{2}={\frac {\hbar }{eB}}} the magnetic length. The energies are:
And the wavefunctions for the motion in the x y {\displaystyle xy} plane are given by the product of a plane wave in y {\displaystyle y} and Hermite polynomials attenuated by the gaussian function in x {\displaystyle x} , which are the wavefunctions of a harmonic oscillator.
From the expression for the Landau levels one notices that the energy depends only on n x {\displaystyle n_{x}} , not on k y {\displaystyle k_{y}} . States with the same n x {\displaystyle n_{x}} but different k y {\displaystyle k_{y}} are degenerate.
At zero field, the density of states per unit surface for the two-dimensional electron gas taking into account degeneration due to spin is independent of the energy
As the field is turned on, the density of states collapses from the constant to a Dirac comb , a series of Dirac δ {\displaystyle \delta } functions, corresponding to the Landau levels separated Δ ε x y = ℏ ω c {\displaystyle \Delta \varepsilon _{xy}=\hbar \omega _{\rm {c}}} . At finite temperature, however, the Landau levels acquire a width Γ = ℏ τ i {\textstyle \Gamma ={\frac {\hbar }{\tau _{i}}}} being τ i {\displaystyle \tau _{i}} the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is a Gaussian or Lorentzian profile.
Another feature is that the wave functions form parallel strips in the y {\displaystyle y} -direction spaced equally along the x {\displaystyle x} -axis, along the lines of A {\displaystyle \mathbf {A} } . Since there is nothing special about any direction in the x y {\displaystyle xy} -plane if the vector potential was differently chosen one should find circular symmetry.
Given a sample of dimensions L x × L y {\displaystyle L_{x}\times L_{y}} and applying the periodic boundary conditions in the y {\displaystyle y} -direction k = 2 π L y j {\textstyle k={\frac {2\pi }{L_{y}}}j} being j {\displaystyle j} an integer, one gets that each parabolic potential is placed at a value x k = l B 2 k {\displaystyle x_{k}=l_{B}^{2}k} .
The number of states for each Landau Level and k {\displaystyle k} can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state.
Thus the density of states per unit surface is
Note the dependency of the density of states with the magnetic field. The larger the magnetic field is, the more states are in each Landau level. As a consequence, there is more confinement in the system since fewer energy levels are occupied.
Rewriting the last expression as n B = ℏ ω c 2 m ∗ π ℏ 2 {\textstyle n_{B}={\frac {\hbar \omega _{\rm {c}}}{2}}{\frac {m^{*}}{\pi \hbar ^{2}}}} it is clear that each Landau level contains as many states as in a 2DEG in a Δ ε = ℏ ω c {\displaystyle \Delta \varepsilon =\hbar \omega _{\rm {c}}} .
Given the fact that electrons are fermions , for each state available in the Landau levels it corresponds to two electrons, one electron with each value for the spin s = ± 1 2 {\textstyle s=\pm {\frac {1}{2}}} . However, if a large magnetic field is applied, the energies split into two levels due to the magnetic moment associated with the alignment of the spin with the magnetic field. The difference in the energies is Δ E = ± 1 2 g μ B B {\textstyle \Delta E=\pm {\frac {1}{2}}g\mu _{\rm {B}}B} being g {\displaystyle g} a factor which depends on the material ( g = 2 {\displaystyle g=2} for free electrons) and μ B {\displaystyle \mu _{\rm {B}}} the Bohr magneton . The sign + {\displaystyle +} is taken when the spin is parallel to the field and − {\displaystyle -} when it is antiparallel. This fact called spin splitting implies that the density of states for each level is reduced by a half. Note that Δ E {\displaystyle \Delta E} is proportional to the magnetic field so, the larger the magnetic field is, the more relevant is the split.
In order to get the number of occupied Landau levels, one defines the so-called filling factor ν {\displaystyle \nu } as the ratio between the density of states in a 2DEG and the density of states in the Landau levels.
In general the filling factor ν {\displaystyle \nu } is not an integer. It happens to be an integer when there is an exact number of filled Landau levels. Instead, it becomes a non-integer when the top level is not fully occupied. In actual experiments, one varies the magnetic field and fixes electron density (and not the Fermi energy!) or varies the electron density and fixes the magnetic field. Both cases correspond to a continuous variation of the filling factor ν {\displaystyle \nu } and one cannot expect ν {\displaystyle \nu } to be an integer. Since n B ∝ B {\displaystyle n_{B}\propto B} , by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level ( ν < 1 {\displaystyle \nu <1} ) and this is called the magnetic quantum limit.
It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When ν {\displaystyle \nu } is an integer, the Fermi energy lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic field is big enough so that there is no overlap between Landau levels, otherwise there would be few electrons and the conductivity would be approximately 0 {\displaystyle 0} ). Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional). [ 24 ]
With the conductivity σ = ρ − 1 {\displaystyle \sigma =\rho ^{-1}} one finds
If the longitudinal resistivity is zero and transversal is finite, then det ρ ≠ 0 {\displaystyle \det \rho \neq 0} . Thus both the longitudinal conductivity and resistivity become zero.
Instead, when ν {\displaystyle \nu } is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the conductivity will have a maximum .
This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called Shubnikov–de Haas oscillations which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since the density of states increases with the field, so there are more carriers which contribute to the resistivity. It is interesting to notice that if the magnetic field is very small, the longitudinal resistivity is a constant which means that the classical result is reached.
From the classical relation of the transverse resistivity ρ x y = B e n 2 D {\textstyle \rho _{xy}={\frac {B}{en_{\rm {2D}}}}} and substituting n 2 D = ν e B h {\textstyle n_{\rm {2D}}=\nu {\frac {eB}{h}}} one finds out the quantization of the transverse resistivity and conductivity:
One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantum e 2 / h {\displaystyle e^{2}/h} if the filling factor is an integer. In experiments, however, plateaus are observed for whole plateaus of filling values ν {\displaystyle \nu } , which indicates that there are in fact electron states between the Landau levels. These states are localized in, for example, impurities of the material where they are trapped in orbits so they can not contribute to the conductivity. That is why the resistivity remains constant in between Landau levels. Again if the magnetic field decreases, one gets the classical result in which the resistivity is proportional to the magnetic field.
The quantum Hall effect, in addition to being observed in two-dimensional electron systems , can be observed in photons. Photons do not possess inherent electric charge , but through the manipulation of discrete optical resonators and coupling phases or on-site phases, an artificial magnetic field can be created. [ 25 ] [ 26 ] [ 27 ] [ 28 ] [ 29 ] This process can be expressed through a metaphor of photons bouncing between multiple mirrors. By shooting the light across multiple mirrors, the photons are routed and gain additional phase proportional to their angular momentum . This creates an effect like they are in a magnetic field .
The integers that appear in the Hall effect are examples of topological quantum numbers . They are known in mathematics as the first Chern numbers and are closely related to Berry's phase . A striking model of much interest in this context is the Azbel–Harper–Hofstadter model whose quantum phase diagram is the Hofstadter butterfly shown in the figure. The vertical axis is the strength of the magnetic field and the horizontal axis is the chemical potential , which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity . In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. Also, the experiments control the filling factor and not the Fermi energy. If this diagram is plotted as a function of filling factor, all the features are completely washed away, hence, it has very little to do with the actual Hall physics.
Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the fractional quantum Hall effect . The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called composite fermions .
The value of the von Klitzing constant may be obtained already on the level of a single atom within the Bohr model while looking at it as a single-electron Hall effect. While during the cyclotron motion on a circular orbit the centrifugal force is balanced by the Lorentz force responsible for the transverse induced voltage and the Hall effect, one may look at the Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle as a Hall current. Defining the single atom Hall current as a rate a single electron charge e {\displaystyle e} is making Kepler revolutions with angular frequency ω {\displaystyle \omega }
and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity:
One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as
which for the Bohr atom is linear but not inverse in the integer n .
Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory . [ 30 ] [ 31 ] | https://en.wikipedia.org/wiki/Quantum_Hall_effect |
Quantum Hall transitions are the quantum phase transitions that occur between different robustly quantized electronic phases of the quantum Hall effect . The robust quantization of these electronic phases is due to strong localization of electrons in their disordered, two-dimensional potential. But, at the quantum Hall transition, the electron gas delocalizes as can be observed in the laboratory. This phenomenon is understood in the language of topological field theory . Here, a vacuum angle (or 'theta angle') distinguishes between topologically different sectors in the vacuum. These topological sectors correspond to the robustly quantized phases. The quantum Hall transitions can then be understood by looking at the topological excitations ( instantons ) that occur between those phases.
Just after the first measurements on the quantum Hall effect in 1980, [ 1 ] physicists wondered how the strongly localized electrons in the disordered potential were able to delocalize at their phase transitions. At that time, the field theory of Anderson localization didn't yet include a topological angle and hence it predicted that: "for any given amount of disorder, all states in two dimensions are localized". A result that was irreconcilable with the observations on delocalization. [ 2 ] Without knowing the solution to this problem, physicists resorted to a semi-classical picture of localized electrons that, given a certain energy, were able to percolate through the disorder. [ 3 ] This percolation mechanism was what assumed to delocalize the electrons
As a result of this semi-classical idea, many numerical computations were done based on the percolation picture. [ 4 ] On top of the classical percolation phase transition, quantum tunneling was included in computer simulations to calculate the critical exponent of the `semi-classical percolation phase transition'. To compare this result with the measured critical exponent, the Fermi-liquid approximation was used, where the Coulomb interactions between electrons are assumed to be finite . Under this assumption, the ground state of the free electron gas can be adiabatically transformed into the ground state of the interacting system and this gives rise to an inelastic scattering length so that the canonical correlation length exponent can be compared to the measured critical exponent.
But, at the quantum phase transition, the localization lengths of the electrons becomes infinite (i.e. they delocalize) and this compromises the Fermi-liquid assumption of an inherently free electron gas (where individual electrons must be well-distinguished). The quantum Hall transition will therefore not be in the Fermi-liquid universality class, but in the ' F -invariant' universality class that has a different value for the critical exponent. [ 5 ] The semi-classical percolation picture of the quantum Hall transition is therefore outdated (although still widely used) and we need to understand the delocalization mechanism as an instanton effect.
The random disorder in the potential landscape of the two-dimensional electron gas plays a key role in the observation of topological sectors and their instantons (phase transitions). Because of the disorder, the electrons are localized and thus they cannot flow across the sample. But if we consider a loop around a localized 2D electron, we can notice that current is still able to flow in the direction around this loop. This current is able to renormalize to larger scales and eventually becomes the Hall current that rotates along the edge of the sample. A topological sector corresponds to an integer number of rotations and it is now visible macroscopically, in the robustly quantized behavior of the measurable Hall current. If the electrons were not sufficiently localized, this measurement would be blurred out by the usual flow of current through the sample.
For the subtle observations on phase transitions it is important that the disorder is of the right kind. The random nature of the potential landscape should be apparent on a scale sufficiently smaller than the sample size in order to clearly distinguish the different phases of the system. These phases are only observable by the principle of emergence, so the difference between self-similar scales has to be multiple orders of magnitude for the critical exponent to be well-defined. On the opposite side, when the disorder correlation length is too small, the states are not sufficiently localized to observe them delocalize.
On the basis of the Renormalization Group Theory of the instanton vacuum one can form a general flow diagram where the topological sectors are represented by attractive fixed points. When scaling the effective system to larger sizes, the system generally flows to a stable phase at one of these points and as we can see in the flow diagram on the right, the longitudinal conductivity will vanish and the Hall conductivity takes on a quantized value. If we started with a Hall conductivity that is halfway between two attractive points, we would end up on the phase transition between topological sectors. As long as the symmetry isn't broken, the longitudinal conductivity doesn't vanish and is even able to increase when scaling to a larger system size. In the flow diagram, we see fixed points that are repulsive in the direction of the Hall current and attractive in the direction of the longitudinal current. It is most interesting to approach these fixed saddle points as close as possible and measure the ( universal ) behavior of the quantum Hall transitions.
If the system is rescaled, the change in conductivity depends only on the distance between a fixed saddle point and the conductivity. The scaling behavior near the quantum Hall transitions is then universal and different quantum Hall samples will give the same scaling results. But, by studying the quantum Hall transitions theoretically, many different systems that are all in different universality classes have been found to share a super-universal fixed point structure. [ 6 ] This means that many different systems that are all in different universality classes still share the same fixed point structure. They all have stable topological sectors and also share other super-universal features. That these features are super-universal is due to the fundamental nature of the vacuum angle that governs the scaling behavior of the systems. The topological vacuum angle can be constructed in any quantum field theory but only under the right circumstances can its features be observed. The vacuum angle also appears in quantum chromodynamics and might have been important in the formation of the early universe. | https://en.wikipedia.org/wiki/Quantum_Hall_transitions |
The quantum Heisenberg model , developed by Werner Heisenberg , is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically . It is related to the prototypical Ising model , where at each site of a lattice, a spin σ i ∈ { ± 1 } {\displaystyle \sigma _{i}\in \{\pm 1\}} represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction .
For quantum mechanical reasons (see exchange interaction or Magnetism § Quantum-mechanical origin of magnetism ), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are aligned . Under this assumption (so that magnetic interactions only occur between adjacent dipoles) and on a 1-dimensional periodic lattice, the Hamiltonian can be written in the form
where J {\displaystyle J} is the coupling constant and dipoles are represented by classical vectors (or "spins") σ j , subject to the periodic boundary condition σ N + 1 = σ 1 {\displaystyle \sigma _{N+1}=\sigma _{1}} .
The Heisenberg model is a more realistic model in that it treats the spins quantum-mechanically, by replacing the spin by a quantum operator acting upon the tensor product ( C 2 ) ⊗ N {\displaystyle (\mathbb {C} ^{2})^{\otimes N}} , of dimension 2 N {\displaystyle 2^{N}} . To define it, recall the Pauli spin-1/2 matrices
and for 1 ≤ j ≤ N {\displaystyle 1\leq j\leq N} and a ∈ { x , y , z } {\displaystyle a\in \{x,y,z\}} denote σ j a = I ⊗ j − 1 ⊗ σ a ⊗ I ⊗ N − j {\displaystyle \sigma _{j}^{a}=I^{\otimes j-1}\otimes \sigma ^{a}\otimes I^{\otimes N-j}} , where I {\displaystyle I} is the 2 × 2 {\displaystyle 2\times 2} identity matrix.
Given a choice of real-valued coupling constants J x , J y , {\displaystyle J_{x},J_{y},} and J z {\displaystyle J_{z}} , the Hamiltonian is given by
where the h {\displaystyle h} on the right-hand side indicates the external magnetic field , with periodic boundary conditions . The objective is to determine the spectrum of the Hamiltonian, from which the partition function can be calculated and the thermodynamics of the system can be studied.
It is common to name the model depending on the values of J x {\displaystyle J_{x}} , J y {\displaystyle J_{y}} and J z {\displaystyle J_{z}} : if J x ≠ J y ≠ J z {\displaystyle J_{x}\neq J_{y}\neq J_{z}} , the model is called the Heisenberg XYZ model; in the case of J = J x = J y ≠ J z = Δ {\displaystyle J=J_{x}=J_{y}\neq J_{z}=\Delta } , it is the Heisenberg XXZ model; if J x = J y = J z = J {\displaystyle J_{x}=J_{y}=J_{z}=J} , it is the Heisenberg XXX model. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz . [ 1 ] In the algebraic formulation, these are related to particular quantum affine algebras and elliptic quantum groups in the XXZ and XYZ cases respectively. [ 2 ] Other approaches do so without Bethe ansatz. [ 3 ]
The physics of the Heisenberg XXX model strongly depends on the sign of the coupling constant J {\displaystyle J} and the dimension of the space. For positive J {\displaystyle J} the ground state is always ferromagnetic . At negative J {\displaystyle J} the ground state is antiferromagnetic in two and three dimensions. [ 4 ] In one dimension the nature of correlations in the antiferromagnetic Heisenberg model depends on the spin of the magnetic dipoles. If the spin is integer then only short-range order is present. A system of half-integer spins exhibits quasi-long range order .
A simplified version of Heisenberg model is the one-dimensional Ising model, where the transverse magnetic field is in the x -direction, and the interaction is only in the z -direction:
At small g and large g , the ground state degeneracy is different, which implies that there must be a quantum phase transition in between. It can be solved exactly for the critical point using the duality analysis. [ 5 ] The duality transition of the Pauli matrices is σ i z = ∏ j ≤ i S j x {\textstyle \sigma _{i}^{z}=\prod _{j\leq i}S_{j}^{x}} and σ i x = S i z S i + 1 z {\displaystyle \sigma _{i}^{x}=S_{i}^{z}S_{i+1}^{z}} , where S x {\displaystyle S^{x}} and S z {\displaystyle S^{z}} are also Pauli matrices which obey the Pauli matrix algebra.
Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form:
but for the g {\displaystyle g} attached to the spin interaction term. Assuming that there's only one critical point, we can conclude that the phase transition happens at g = 1 {\displaystyle g=1} .
Following the approach of Ludwig Faddeev ( 1996 ), the spectrum of the Hamiltonian for the XXX model H = 1 4 ∑ α , n ( σ n α σ n + 1 α − 1 ) {\displaystyle H={\frac {1}{4}}\sum _{\alpha ,n}(\sigma _{n}^{\alpha }\sigma _{n+1}^{\alpha }-1)} can be determined by the Bethe ansatz. In this context, for an appropriately defined family of operators B ( λ ) {\displaystyle B(\lambda )} dependent on a spectral parameter λ ∈ C {\displaystyle \lambda \in \mathbb {C} } acting on the total Hilbert space H = ⨂ n = 1 N h n {\displaystyle {\mathcal {H}}=\bigotimes _{n=1}^{N}h_{n}} with each h n ≅ C 2 {\displaystyle h_{n}\cong \mathbb {C} ^{2}} , a Bethe vector is a vector of the form Φ ( λ 1 , ⋯ , λ m ) = B ( λ 1 ) ⋯ B ( λ m ) v 0 {\displaystyle \Phi (\lambda _{1},\cdots ,\lambda _{m})=B(\lambda _{1})\cdots B(\lambda _{m})v_{0}} where v 0 = ⨂ n = 1 N | ↑ ⟩ {\displaystyle v_{0}=\bigotimes _{n=1}^{N}|\uparrow \,\rangle } .
If the λ k {\displaystyle \lambda _{k}} satisfy the Bethe equation ( λ k + i / 2 λ k − i / 2 ) N = ∏ j ≠ k λ k − λ j + i λ k − λ j − i , {\displaystyle \left({\frac {\lambda _{k}+i/2}{\lambda _{k}-i/2}}\right)^{N}=\prod _{j\neq k}{\frac {\lambda _{k}-\lambda _{j}+i}{\lambda _{k}-\lambda _{j}-i}},} then the Bethe vector is an eigenvector of H {\displaystyle H} with eigenvalue − ∑ k 1 2 1 λ k 2 + 1 / 4 {\displaystyle -\sum _{k}{\frac {1}{2}}{\frac {1}{\lambda _{k}^{2}+1/4}}} .
The family B ( λ ) {\displaystyle B(\lambda )} as well as three other families come from a transfer matrix T ( λ ) {\displaystyle T(\lambda )} (in turn defined using a Lax matrix ), which acts on H {\displaystyle {\mathcal {H}}} along with an auxiliary space h a ≅ C 2 {\displaystyle h_{a}\cong \mathbb {C} ^{2}} , and can be written as a 2 × 2 {\displaystyle 2\times 2} block matrix with entries in E n d ( H ) {\displaystyle \mathrm {End} ({\mathcal {H}})} , T ( λ ) = ( A ( λ ) B ( λ ) C ( λ ) D ( λ ) ) , {\displaystyle T(\lambda )={\begin{pmatrix}A(\lambda )&B(\lambda )\\C(\lambda )&D(\lambda )\end{pmatrix}},} which satisfies fundamental commutation relations (FCRs) similar in form to the Yang–Baxter equation used to derive the Bethe equations. The FCRs also show there is a large commuting subalgebra given by the generating function F ( λ ) = t r a ( T ( λ ) ) = A ( λ ) + D ( λ ) {\displaystyle F(\lambda )=\mathrm {tr} _{a}(T(\lambda ))=A(\lambda )+D(\lambda )} , as [ F ( λ ) , F ( μ ) ] = 0 {\displaystyle [F(\lambda ),F(\mu )]=0} , so when F ( λ ) {\displaystyle F(\lambda )} is written as a polynomial in λ {\displaystyle \lambda } , the coefficients all commute, spanning a commutative subalgebra which H {\displaystyle H} is an element of. The Bethe vectors are in fact simultaneous eigenvectors for the whole subalgebra.
For higher spins, say spin s {\displaystyle s} , replace σ α {\displaystyle \sigma ^{\alpha }} with S α {\displaystyle S^{\alpha }} coming from the Lie algebra representation of the Lie algebra s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} , of dimension 2 s + 1 {\displaystyle 2s+1} . The XXX s Hamiltonian H = ∑ α , n ( S n α S n + 1 α − ( S n α S n + 1 α ) 2 ) {\displaystyle H=\sum _{\alpha ,n}(S_{n}^{\alpha }S_{n+1}^{\alpha }-(S_{n}^{\alpha }S_{n+1}^{\alpha })^{2})} is solvable by Bethe ansatz with Bethe equations ( λ k + i s λ k − i s ) N = ∏ j ≠ k λ k − λ j + i λ k − λ j − i . {\displaystyle \left({\frac {\lambda _{k}+is}{\lambda _{k}-is}}\right)^{N}=\prod _{j\neq k}{\frac {\lambda _{k}-\lambda _{j}+i}{\lambda _{k}-\lambda _{j}-i}}.}
For spin s {\displaystyle s} and a parameter γ {\displaystyle \gamma } for the deformation from the XXX model, the BAE (Bethe ansatz equation) is ( sinh ( λ k + i s γ ) sinh ( λ k − i s γ ) ) N = ∏ j ≠ k sinh ( λ k − λ j + i γ ) sinh ( λ k − λ j − i γ ) . {\displaystyle \left({\frac {\sinh(\lambda _{k}+is\gamma )}{\sinh(\lambda _{k}-is\gamma )}}\right)^{N}=\prod _{j\neq k}{\frac {\sinh(\lambda _{k}-\lambda _{j}+i\gamma )}{\sinh(\lambda _{k}-\lambda _{j}-i\gamma )}}.} Notably, for s = 1 2 {\displaystyle s={\frac {1}{2}}} these are precisely the BAEs for the six-vertex model , after identifying γ = 2 η {\displaystyle \gamma =2\eta } , where η {\displaystyle \eta } is the anisotropy parameter of the six-vertex model. [ 6 ] [ 7 ] This was originally thought to be coincidental until Baxter showed the XXZ Hamiltonian was contained in the algebra generated by the transfer matrix T ( ν ) {\displaystyle T(\nu )} , [ 8 ] given exactly by H X X Z 1 / 2 = − i sin 2 η d d ν log T ( ν ) | ν = − i η − 1 2 cos 2 η 1 ⊗ N . {\displaystyle H_{XXZ_{1/2}}=-i\sin 2\eta {\frac {d}{d\nu }}\log T(\nu ){\Big |}_{\nu =-i\eta }-{\frac {1}{2}}\cos 2\eta 1^{\otimes N}.}
The integrability is underpinned by the existence of large symmetry algebras for the different models. For the XXX case this is the Yangian Y ( s l 2 ) {\displaystyle Y({\mathfrak {sl}}_{2})} , while in the XXZ case this is the quantum group s l q ( 2 ) ^ {\displaystyle {\hat {{\mathfrak {sl}}_{q}(2)}}} , the q-deformation of the affine Lie algebra of s l 2 ^ {\displaystyle {\hat {{\mathfrak {sl}}_{2}}}} , as explained in the notes by Faddeev ( 1996 ).
These appear through the transfer matrix, and the condition that the Bethe vectors are generated from a state Ω {\displaystyle \Omega } satisfying C ( λ ) ⋅ Ω = 0 {\displaystyle C(\lambda )\cdot \Omega =0} corresponds to the solutions being part of a highest-weight representation of the extended symmetry algebras. | https://en.wikipedia.org/wiki/Quantum_Heisenberg_model |
In mathematical physics , the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras . They are a consistent system of difference equations satisfied by the N -point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter q approaches 1, the N -point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations . The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics .
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This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_KZ_equations |
In quantum mechanics , a quantum Markov semigroup describes the dynamics in a Markovian open quantum system . The axiomatic definition of the prototype of quantum Markov semigroups was first introduced by A. M. Kossakowski [ 1 ] in 1972, and then developed by V. Gorini, A. M. Kossakowski , E. C. G. Sudarshan [ 2 ] and Göran Lindblad [ 3 ] in 1976. [ 4 ]
An ideal quantum system is not realistic because it should be completely isolated while, in practice, it is influenced by the coupling to an environment, which typically has a large number of degrees of freedom (for example an atom interacting with the surrounding radiation field). A complete microscopic description of the degrees of freedom of the environment is typically too complicated. Hence, one looks for simpler descriptions of the dynamics of the open system. In principle, one should investigate the unitary dynamics of the total system, i.e. the system and the environment, to obtain information about the reduced system of interest by averaging the appropriate observables over the degrees of freedom of the environment. To model the dissipative effects due to the interaction with the environment, the Schrödinger equation is replaced by a suitable master equation , such as a Lindblad equation or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few quantum noises . Mathematically, time evolution in a Markovian open quantum system is no longer described by means of one-parameter groups of unitary maps, but one needs to introduce quantum Markov semigroups .
In general, quantum dynamical semigroups can be defined on von Neumann algebras , so the dimensionality of the system could be infinite. Let A {\displaystyle {\mathcal {A}}} be a von Neumann algebra acting on Hilbert space H {\displaystyle {\mathcal {H}}} , a quantum dynamical semigroup on A {\displaystyle {\mathcal {A}}} is a collection of bounded operators on A {\displaystyle {\mathcal {A}}} , denoted by T := ( T t ) t ≥ 0 {\displaystyle {\mathcal {T}}:=\left({\mathcal {T}}_{t}\right)_{t\geq 0}} , with the following properties: [ 5 ]
Under the condition of complete positivity, the operators T t {\displaystyle {\mathcal {T}}_{t}} are σ {\displaystyle \sigma } -weakly continuous if and only if T t {\displaystyle {\mathcal {T}}_{t}} are normal. [ 5 ] Recall that, letting A + {\displaystyle {\mathcal {A}}_{+}} denote the convex cone of positive elements in A {\displaystyle {\mathcal {A}}} , a positive operator T : A → A {\displaystyle T:{\mathcal {A}}\rightarrow {\mathcal {A}}} is said to be normal if for every increasing net ( x α ) α {\displaystyle \left(x_{\alpha }\right)_{\alpha }} in A + {\displaystyle {\mathcal {A}}_{+}} with least upper bound x {\displaystyle x} in A + {\displaystyle {\mathcal {A}}_{+}} one has
for each u {\displaystyle u} in a norm-dense linear sub-manifold of H {\displaystyle {\mathcal {H}}} .
A quantum dynamical semigroup T {\displaystyle {\mathcal {T}}} is said to be identity-preserving (or conservative, or Markovian) if
where 1 ∈ A {\displaystyle {\boldsymbol {1}}\in {\mathcal {A}}} is the identity element. For simplicity, T {\displaystyle {\mathcal {T}}} is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of T t {\displaystyle {\mathcal {T}}_{t}} imply ‖ T t ‖ = 1 {\displaystyle \left\|{\mathcal {T}}_{t}\right\|=1} for all t ≥ 0 {\displaystyle t\geq 0} and then T {\displaystyle {\mathcal {T}}} is a contraction semigroup . [ 6 ]
The Condition ( 1 ) plays an important role not only in the proof of uniqueness and unitarity of solution of a Hudson – Parthasarathy quantum stochastic differential equation , but also in deducing regularity conditions for paths of classical Markov processes in view of operator theory . [ 7 ]
The infinitesimal generator of a quantum dynamical semigroup T {\displaystyle {\mathcal {T}}} is the operator L {\displaystyle {\mathcal {L}}} with domain Dom ( L ) {\displaystyle \operatorname {Dom} ({\mathcal {L}})} , where
and L ( a ) := b {\displaystyle {\mathcal {L}}(a):=b} .
If the quantum Markov semigroup T {\displaystyle {\mathcal {T}}} is uniformly continuous in addition, which means lim t → 0 + ‖ T t − T 0 ‖ = 0 {\displaystyle \lim _{t\rightarrow 0^{+}}\left\|{\mathcal {T}}_{t}-{\mathcal {T}}_{0}\right\|=0} , then
Under such assumption, the infinitesimal generator L {\displaystyle {\mathcal {L}}} has the characterization [ 3 ]
where a ∈ A {\displaystyle a\in {\mathcal {A}}} , V j ∈ B ( H ) {\displaystyle V_{j}\in {\mathcal {B}}({\mathcal {H}})} , ∑ j V j † V j ∈ B ( H ) {\displaystyle \sum _{j}V_{j}^{\dagger }V_{j}\in {\mathcal {B}}({\mathcal {H}})} , and H ∈ B ( H ) {\displaystyle H\in {\mathcal {B}}({\mathcal {H}})} is self-adjoint . Moreover, above [ ⋅ , ⋅ ] {\displaystyle \left[\cdot ,\cdot \right]} denotes the commutator , and { ⋅ , ⋅ } {\displaystyle \left\{\cdot ,\cdot \right\}} the anti-commutator . | https://en.wikipedia.org/wiki/Quantum_Markov_semigroup |
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems . One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantum many-body problem . The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem.
Quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in the wave function , going beyond mean-field theory . In particular, there exist numerically exact and polynomially -scaling algorithms to exactly study static properties of boson systems without geometrical frustration . For fermions , there exist very good approximations to their static properties and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both.
In principle, any physical system can be described by the many-body Schrödinger equation as long as the constituent particles are not moving "too" fast; that is, they are not moving at a speed comparable to that of light, and relativistic effects can be neglected. This is true for a wide range of electronic problems in condensed matter physics , in Bose–Einstein condensates and superfluids such as liquid helium . The ability to solve the Schrödinger equation for a given system allows prediction of its behavior, with important applications ranging from materials science to complex biological systems .
The difficulty is however that solving the Schrödinger equation requires the knowledge of the many-body wave function in the many-body Hilbert space , which typically has an exponentially large size in the number of particles. Its solution for a reasonably large number of particles is therefore typically impossible, even for modern parallel computing technology in a reasonable amount of time. Traditionally, approximations for the many-body wave function as an antisymmetric function of one-body orbitals [ 1 ] have been used, in order to have a manageable treatment of the Schrödinger equation. However, this kind of formulation has several drawbacks, either limiting the effect of quantum many-body correlations, as in the case of the Hartree–Fock (HF) approximation, or converging very slowly, as in configuration interaction applications in quantum chemistry.
Quantum Monte Carlo is a way to directly study the many-body problem and the many-body wave function beyond these approximations. The most advanced quantum Monte Carlo approaches provide an exact solution to the many-body problem for non-frustrated interacting boson systems, while providing an approximate description of interacting fermion systems. Most methods aim at computing the ground state wavefunction of the system, with the exception of path integral Monte Carlo and finite-temperature auxiliary-field Monte Carlo , which calculate the density matrix . In addition to static properties, the time-dependent Schrödinger equation can also be solved, albeit only approximately, restricting the functional form of the time-evolved wave function , as done in the time-dependent variational Monte Carlo .
From a probabilistic point of view, the computation of the top eigenvalues and the corresponding ground state eigenfunctions associated with the Schrödinger equation relies on the numerical solving of Feynman–Kac path integration problems. [ 2 ] [ 3 ]
There are several quantum Monte Carlo methods, each of which uses Monte Carlo in different ways to solve the many-body problem. | https://en.wikipedia.org/wiki/Quantum_Monte_Carlo |
Quantum Trajectory Theory (QTT) is a formulation of quantum mechanics used for simulating open quantum systems , quantum dissipation and single quantum systems. [ 1 ] It was developed by Howard Carmichael in the early 1990s around the same time as the similar formulation, known as the quantum jump method or Monte Carlo wave function (MCWF) method, developed by Dalibard , Castin and Mølmer . [ 2 ] Other contemporaneous works on wave-function-based Monte Carlo approaches to open quantum systems include those of Dum, Zoller and Ritsch , and Hegerfeldt and Wilser. [ 3 ]
QTT is compatible with the standard formulation of quantum theory, as described by the Schrödinger equation , but it offers a more detailed view. [ 4 ] [ 1 ] The Schrödinger equation can be used to compute the probability of finding a quantum system in each of its possible states should a measurement be made. This approach is fundamentally statistical and is useful for predicting average measurements of large ensembles of quantum objects but it does not describe or provide insight into the behaviour of individual particles. QTT fills this gap by offering a way to describe the trajectories of individual quantum particles that obey the probabilities computed from the Schrödinger equation. [ 4 ] [ 5 ] Like the quantum jump method, QTT applies to open quantum systems that interact with their environment. [ 1 ] QTT has become particularly popular since the technology has been developed to efficiently control and monitor individual quantum systems as it can predict how individual quantum objects such as particles will behave when they are observed. [ 4 ]
In QTT open quantum systems are modelled as scattering processes, with classical external fields corresponding to the inputs and classical stochastic processes corresponding to the outputs (the fields after the measurement process). [ 6 ] The mapping from inputs to outputs is provided by a quantum stochastic process that is set up to account for a particular measurement strategy (e.g., photon counting , homodyne / heterodyne detection, etc.). [ 7 ] The calculated system state as a function of time is known as a quantum trajectory , and the desired density matrix as a function of time may be calculated by averaging over many simulated trajectories.
Like other Monte Carlo approaches, QTT provides an advantage over direct master-equation approaches by reducing the number of computations required. For a Hilbert space of dimension N, the traditional master equation approach would require calculation of the evolution of N 2 atomic density matrix elements, whereas QTT only requires N calculations. This makes it useful for simulating large open quantum systems. [ 8 ]
The idea of monitoring outputs and building measurement records is fundamental to QTT. This focus on measurement distinguishes it from the quantum jump method which has no direct connection to monitoring output fields. When applied to direct photon detection the two theories produce equivalent results. Where the quantum jump method predicts the quantum jumps of the system as photons are emitted, QTT predicts the "clicks" of the detector as photons are measured. The only difference is the viewpoint. [ 8 ]
QTT is also broader in its application than the quantum jump method as it can be applied to many different monitoring strategies including direct photon detection and heterodyne detection. Each different monitoring strategy offers a different picture of the system dynamics. [ 8 ]
There have been two distinct phases of applications for QTT. Like the quantum jump method, QTT was first used for computer simulations of large quantum systems. These applications exploit its ability to significantly reduce the size of computations, which was especially necessary in the 1990s when computing power was very limited. [ 2 ] [ 9 ] [ 10 ]
The second phase of application has been catalysed by the development of technologies to precisely control and monitor single quantum systems. In this context QTT is being used to predict and guide single quantum system experiments including those contributing to the development of quantum computers. [ 1 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 5 ]
It has also been shown that quantum trajectories have full and universal quantum computational power. [ 16 ]
QTT addresses one aspect of the measurement problem in quantum mechanics by providing a detailed description of the intermediate steps through which a quantum state approaches the final, measured state during the so-called " collapse of the wave function ". It reconciles the concept of a quantum jump with the smooth evolution described by the Schrödinger equation . The theory suggests that "quantum jumps" are not instantaneous but happen in a coherently driven system as a smooth transition through a series of superposition states . [ 5 ] This prediction was tested experimentally in 2019 by a team at Yale University led by Michel Devoret and Zlatko Minev , in collaboration with Carmichael and others at Yale University and the University of Auckland . In their experiment they used a superconducting artificial atom to observe a quantum jump in detail, confirming that the transition is a continuous process that unfolds over time. They were also able to detect when a quantum jump was about to occur and intervene to reverse it, sending the system back to the state in which it started. [ 11 ] This experiment, inspired and guided by QTT, represents a new level of control over quantum systems and has potential applications in correcting errors in quantum computing in the future. [ 11 ] [ 17 ] [ 18 ] [ 19 ] [ 5 ] [ 1 ] | https://en.wikipedia.org/wiki/Quantum_Trajectory_Theory |
In physics , quantum acoustics is the study of sound under conditions such that quantum mechanical effects are relevant. For most applications, classical mechanics are sufficient to accurately describe the physics of sound. However very high frequency sounds, or sounds made at very low temperatures may be subject to quantum effects.
Quantum acoustics [ 1 ] can also refer to attempts within the scientific community to couple superconducting qubits to acoustic waves. [ 2 ] One particularly successful method involves coupling a superconducting qubit with a Surface Acoustic Wave (SAW) Resonator and placing these components on different substrates to achieve a higher signal to noise ratio as well as controlling the coupling strength of the components. This allows quantum experiments to verify that the phonons within the SAW Resonator are in quantum fock states by using Quantum tomography . [ 3 ] Similar attempts have been made by using bulk acoustic resonators. [ 4 ] One consequence of these developments is that it is possible to explore the properties of atoms with a much larger size than found conventionally by modelling them using a superconducting qubit coupled with a SAW Resonator. [ 5 ]
Most recently, quantum acoustics has been used as a term to describe the coherent state limit of lattice vibrations , in analogue to quantum optics . [ 6 ] | https://en.wikipedia.org/wiki/Quantum_acoustics |
Quantum algebra is one of the top-level mathematics categories used by the arXiv . It is the study of noncommutative analogues and generalizations of commutative algebras, especially those arising in Lie theory . [ 1 ]
Subjects include:
This algebra -related article is a stub . You can help Wikipedia by expanding it .
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it .
This mathematical physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_algebra |
In physics , a quantum amplifier is an amplifier that uses quantum mechanical methods to amplify a signal; examples include the active elements of lasers and optical amplifiers .
The main properties of the quantum amplifier are its amplification coefficient and uncertainty . These parameters are not independent; the higher the amplification coefficient, the higher the uncertainty (noise). In the case of lasers, the uncertainty corresponds to the amplified spontaneous emission of the active medium. The unavoidable noise of quantum amplifiers is one of the reasons for the use of digital signals in optical communications and can be deduced from the fundamentals of quantum mechanics.
An amplifier increases the amplitude of whatever goes through it. While classical amplifiers take in classical signals, quantum amplifiers take in quantum signals, such as coherent states . This does not necessarily mean that the output is a coherent state; indeed, typically it is not. The form of the output depends on the specific amplifier design. Besides amplifying the intensity of the input, quantum amplifiers can also increase the quantum noise present in the signal.
The physical electric field in a paraxial single-mode pulse can be approximated with superposition of modes; the electric field E p h y s {\displaystyle ~E_{\rm {phys}}~} of a single mode can be described as
where
The analysis of the noise in the system is made with respect to the mean value [ clarification needed ] of the annihilation operator. To obtain the noise, one solves for the real and imaginary parts of the projection of the field to a given mode M ( x → ) {\displaystyle ~M({\vec {x}})~} . Spatial coordinates do not appear in the solution.
Assume that the mean value of the initial field is ⟨ a ^ ⟩ i n i t i a l {\displaystyle ~{\left\langle {\hat {a}}\right\rangle _{\rm {initial}}}~} . Physically, the initial state corresponds to the coherent pulse at the input of the optical amplifier; the final state corresponds to the output pulse. The amplitude-phase behavior of the pulse must be known, although only the quantum state of the corresponding mode is important. The pulse may be treated in terms of a single-mode field.
A quantum amplifier is a unitary transform U ^ {\displaystyle {\hat {U}}} , acting the initial state | i n i t i a l ⟩ {\displaystyle ~|{\rm {initial}}\rangle ~} and producing the amplified state | f i n a l ⟩ {\displaystyle ~|{\rm {final}}\rangle ~} , as follows:
This equation describes the quantum amplifier in the Schrödinger representation .
The amplification depends on the mean value ⟨ a ^ ⟩ {\displaystyle ~\langle {\hat {a}}\rangle ~} of the field operator a ^ {\displaystyle ~{\hat {a}}~} and its dispersion ⟨ a ^ † a ^ ⟩ − ⟨ a ^ † ⟩ ⟨ a ^ ⟩ {\displaystyle ~\langle {\hat {a}}^{\dagger }{\hat {a}}\rangle -\langle {\hat {a}}^{\dagger }\rangle \langle {\hat {a}}\rangle ~} . A coherent state is a state with minimal uncertainty; when the state is transformed, the uncertainty may increase. This increase can be interpreted as noise in the amplifier.
The gain G {\displaystyle ~G~} can be defined as follows:
The can be written also in the Heisenberg representation ; the changes are attributed to the amplification of the field operator. Thus, the evolution of the operator A is given by A ^ = U ^ † a ^ U ^ {\displaystyle ~{\hat {A}}={\hat {U}}^{\dagger }{\hat {a}}{\hat {U}}~} , while the state vector remains unchanged. The gain is given by
In general, the gain G {\displaystyle ~G~} may be complex, and it may depend on the initial state. For laser applications, the amplification of coherent states is important. Therefore, it is usually assumed that the initial state is a coherent state characterized by a complex-valued initial parameter α {\displaystyle ~\alpha ~} such that | i n i t i a l ⟩ = | α ⟩ {\displaystyle ~~|{\rm {initial}}\rangle =|\alpha \rangle ~} . Even with such a restriction, the gain may depend on the amplitude or phase of the initial field.
In the following, the Heisenberg representation is used; all brackets are assumed to be evaluated with respect to the initial coherent state.
The expectation values are assumed to be evaluated with respect to the initial coherent state. This quantity characterizes the increase of the uncertainty of the field due to amplification. As the uncertainty of the field operator does not depend on its parameter, the quantity above shows how much output field differs from a coherent state.
Linear phase-invariant amplifiers may be described as follows. Assume that the unitary operator U ^ {\displaystyle ~{\hat {U}}~} amplifies in such a way that the input a ^ {\displaystyle ~{\hat {a}}~} and the output A ^ = U ^ † a ^ U ^ {\displaystyle ~{\hat {A}}={\hat {U}}^{\dagger }{\hat {a}}{\hat {U}}~} are related by a linear equation
where c {\displaystyle ~c~} and s {\displaystyle ~s~} are c-numbers and b ^ † {\displaystyle ~{\hat {b}}^{\dagger }~} is a creation operator characterizing the amplifier. Without loss of generality, it may be assumed that c {\displaystyle ~c~} and s {\displaystyle ~s~} are real . The commutator of the field operators is invariant under unitary transformation U ^ {\displaystyle ~{\hat {U}}~} :
From the unitarity of U ^ {\displaystyle ~{\hat {U}}~} , it follows that b ^ {\displaystyle ~{\hat {b}}~} satisfies the canonical commutation relations for operators with Bose statistics :
The c-numbers are then
Hence, the phase-invariant amplifier acts by introducing an additional mode to the field, with a large amount of stored energy, behaving as a boson . Calculating the gain and the noise of this amplifier, one finds
and
The coefficient g = | G | 2 {\displaystyle ~~g\!=\!|G|^{2}~~} is sometimes called the intensity amplification coefficient . The noise of the linear phase-invariant amplifier is given by g − 1 {\displaystyle g-1} . The gain can be dropped by splitting the beam; the estimate above gives the minimal possible noise of the linear phase-invariant amplifier.
The linear amplifier has an advantage over the multi-mode amplifier: if several modes of a linear amplifier are amplified by the same factor, the noise in each mode is determined independently;that is, modes in a linear quantum amplifier are independent.
To obtain a large amplification coefficient with minimal noise, one may use homodyne detection , constructing a field state with known amplitude and phase, corresponding to the linear phase-invariant amplifier. [ 2 ] The uncertainty principle sets the lower bound of quantum noise in an amplifier. In particular, the output of a laser system and the output of an optical generator are not coherent states.
Nonlinear amplifiers do not have a linear relation between their input and output. The maximum noise of a nonlinear amplifier cannot be much smaller than that of an idealized linear amplifier. [ 1 ] This limit is determined by the derivatives of the mapping function; a larger derivative implies an amplifier with greater uncertainty. [ 3 ] Examples include most lasers, which include near-linear amplifiers, operating close to their threshold and thus exhibiting large uncertainty and nonlinear operation. As with the linear amplifiers, they may preserve the phase and keep the uncertainty low, but there are exceptions. These include parametric oscillators , which amplify while shifting the phase of the input. | https://en.wikipedia.org/wiki/Quantum_amplifier |
Quantum biology is the study of applications of quantum mechanics and theoretical chemistry to aspects of biology that cannot be accurately described by the classical laws of physics. [ 1 ] An understanding of fundamental quantum interactions is important because they determine the properties of the next level of organization in biological systems.
Many biological processes involve the conversion of energy into forms that are usable for chemical transformations, and are quantum mechanical in nature. Such processes involve chemical reactions , light absorption , formation of excited electronic states , transfer of excitation energy , and the transfer of electrons and protons ( hydrogen ions ) in chemical processes, such as photosynthesis , olfaction and cellular respiration . [ 2 ] Moreover, quantum biology may use computations to model biological interactions in light of quantum mechanical effects. [ 3 ] Quantum biology is concerned with the influence of non-trivial quantum phenomena, [ 4 ] which can be explained by reducing the biological process to fundamental physics , although these effects are difficult to study and can be speculative. [ 5 ]
Currently, there exist four major life processes that have been identified as influenced by quantum effects: enzyme catalysis, sensory processes, energy transference, and information encoding. [ 6 ]
Quantum biology is an emerging field, in the sense that most current research is theoretical and subject to questions that require further experimentation. Though the field has only recently received an influx of attention, it has been conceptualized by physicists throughout the 20th century. It has been suggested that quantum biology might play a critical role in the future of the medical world. [ 7 ] Early pioneers of quantum physics saw applications of quantum mechanics in biological problems. Erwin Schrödinger 's 1944 book What Is Life? discussed applications of quantum mechanics in biology. [ 8 ] Schrödinger introduced the idea of an " aperiodic crystal " that contained genetic information in its configuration of covalent chemical bonds . He further suggested that mutations are introduced by "quantum leaps". Other pioneers Niels Bohr , Pascual Jordan , and Max Delbrück argued that the quantum idea of complementarity was fundamental to the life sciences. [ 9 ] In 1963, Per-Olov Löwdin published proton tunneling as another mechanism for DNA mutation. In his paper, he stated that there is a new field of study called "quantum biology". [ 10 ] In 1979, the Soviet and Ukrainian physicist Alexander Davydov published the first textbook on quantum biology entitled Biology and Quantum Mechanics . [ 11 ] [ 12 ]
Enzymes have been postulated to use quantum tunneling to transfer electrons in electron transport chains . [ 13 ] [ 14 ] [ 15 ] It is possible that protein quaternary architectures may have adapted to enable sustained quantum entanglement and coherence , which are two of the limiting factors for quantum tunneling in biological entities. [ 16 ] These architectures might account for a greater percentage of quantum energy transfer, which occurs through electron transport and proton tunneling (usually in the form of hydrogen ions, H + ). [ 17 ] [ 18 ] Tunneling refers to the ability of a subatomic particle to travel through potential energy barriers. [ 19 ] This ability is due, in part, to the principle of complementarity , which holds that certain substances have pairs of properties that cannot be measured separately without changing the outcome of measurement. Particles, such as electrons and protons, have wave-particle duality; they can pass through energy barriers due to their wave characteristics without violating the laws of physics. In order to quantify how quantum tunneling is used in many enzymatic activities, many biophysicists utilize the observation of hydrogen ions. When hydrogen ions are transferred, this is seen as a staple in an organelle's primary energy processing network; in other words, quantum effects are most usually at work in proton distribution sites at distances on the order of an angstrom (1 Å). [ 20 ] [ 21 ] In physics, a semiclassical (SC) approach is most useful in defining this process because of the transfer from quantum elements (e.g. particles) to macroscopic phenomena (e.g. biochemicals ). Aside from hydrogen tunneling, studies also show that electron transfer between redox centers through quantum tunneling plays an important role in enzymatic activity of photosynthesis and cellular respiration (see also Mitochondria section below). [ 15 ] [ 22 ]
Ferritin is an iron storage protein that is found in plants and animals. It is usually formed from 24 subunits that self-assemble into a spherical shell that is approximately 2 nm thick, with an outer diameter that varies with iron loading up to about 16 nm. Up to ~4500 iron atoms can be stored inside the core of the shell in the Fe3+ oxidation state as water-insoluble compounds such as ferrihydrite and magnetite . [ 23 ] Ferritin is able to store electrons for at least several hours, which reduce the Fe3+ to water soluble Fe2+. [ 24 ] Electron tunneling as the mechanism by which electrons transit the 2 nm thick protein shell was proposed as early as 1988. [ 25 ] Electron tunneling and other quantum mechanical properties of ferritin were observed in 1992, [ 26 ] and electron tunneling at room temperature and ambient conditions was observed in 2005. [ 27 ] Electron tunneling associated with ferritin is a quantum biological process, and ferritin is a quantum biological agent.
Electron tunneling through ferritin between electrodes is independent of temperature, which indicates that it is substantially coherent and activation-less. [ 28 ] The electron tunneling distance is a function of the size of the ferritin. Single electron tunneling events can occur over distances of up to 8 nm through the ferritin, and sequential electron tunneling can occur up to 12 nm through the ferritin. It has been proposed that the electron tunneling is magnon-assisted and associated with magnetite microdomains in the ferritin core. [ 29 ]
Early evidence of quantum mechanical properties exhibited by ferritin in vivo was reported in 2004, where increased magnetic ordering of ferritin structures in placental macrophages was observed using small angle neutron scattering (SANS). [ 30 ] Quantum dot solids also show increased magnetic ordering in SANS testing, [ 31 ] and can conduct electrons over long distances. [ 32 ] Increased magnetic ordering of ferritin cores disposed in an ordered layer on a silicon substrate with SANS testing has also been observed. [ 33 ] Ferritin structures like those in placental macrophages have been tested in solid state configurations and exhibit quantum dot solid-like properties of conducting electrons over distances of up to 80 microns through sequential tunneling and formation of Coulomb blockades. [ 34 ] [ 35 ] [ 36 ] Electron transport through ferritin in placental macrophages may be associated with an anti-inflammatory function. [ 37 ]
Conductive atomic force microscopy of substantia nigra pars compacta (SNc) tissue demonstrated evidence of electron tunneling between ferritin cores, in structures that correlate to layers of ferritin outside of neuromelanin organelles. [ 38 ]
Evidence of ferritin layers in cell bodies of large dopamine neurons of the SNc and between those cell bodies in glial cells has also been found, [ 39 ] [ 40 ] [ 41 ] and is hypothesized to be associated with neuron function. [ 42 ] Overexpression of ferritin reduces the accumulation of reactive oxygen species (ROS), [ 43 ] and may act as a catalyst by increasing the ability of electrons from antioxidants to neutralize ROS through electron tunneling. Ferritin has also been observed in ordered configurations in lysosomes associated with erythropoiesis , [ 44 ] where it may be associated with red blood cell production. While direct evidence of tunneling associated with ferritin in vivo in live cells has not yet been obtained, it may be possible to do so using QDs tagged with anti-ferritin, which should emit photons if electrons stored in the ferritin core tunnel to the QD. [ 45 ]
Olfaction, the sense of smell, can be broken down into two parts; the reception and detection of a chemical, and how that detection is sent to and processed by the brain. This process of detecting an odorant is still under question. One theory named the " shape theory of olfaction " suggests that certain olfactory receptors are triggered by certain shapes of chemicals and those receptors send a specific message to the brain. [ 46 ] Another theory (based on quantum phenomena) suggests that the olfactory receptors detect the vibration of the molecules that reach them and the "smell" is due to different vibrational frequencies, this theory is aptly called the "vibration theory of olfaction."
The vibration theory of olfaction , created in 1938 by Malcolm Dyson [ 47 ] but reinvigorated by Luca Turin in 1996, [ 48 ] proposes that the mechanism for the sense of smell is due to G-protein receptors that detect molecular vibrations due to inelastic electron tunneling, tunneling where the electron loses energy, across molecules. [ 48 ] In this process a molecule would fill a binding site with a G-protein receptor. After the binding of the chemical to the receptor, the chemical would then act as a bridge allowing for the electron to be transferred through the protein. As the electron transfers across what would otherwise have been a barrier, it loses energy due to the vibration of the newly-bound molecule to the receptor. This results in the ability to smell the molecule. [ 48 ] [ 4 ]
While the vibration theory has some experimental proof of concept, [ 49 ] [ 50 ] there have been multiple controversial results in experiments. In some experiments, animals are able to distinguish smells between molecules of different frequencies and same structure, [ 51 ] while other experiments show that people are unaware of distinguishing smells due to distinct molecular frequencies. [ 52 ]
Vision relies on quantized energy in order to convert light signals to an action potential in a process called phototransduction . In phototransduction, a photon interacts with a chromophore in a light receptor. The chromophore absorbs the photon and undergoes photoisomerization . This change in structure induces a change in the structure of the photo receptor and resulting signal transduction pathways lead to a visual signal. However, the photoisomerization reaction occurs at a rapid rate, in under 200 femtoseconds , [ 53 ] with high yield. Models suggest the use of quantum effects in shaping the ground state and excited state potentials in order to achieve this efficiency. [ 54 ]
The sensor in the retina of the human eye is sensitive enough to detect a single photon. [ 55 ] Single photon detection could lead to multiple different technologies. One area of development is in quantum communication and cryptography . The idea is to use a biometric system to measure the eye using only a small number of points across the retina with random flashes of photons that "read" the retina and identify the individual. [ 56 ] This biometric system would only allow a certain individual with a specific retinal map to decode the message. This message can not be decoded by anyone else unless the eavesdropper were to guess the proper map or could read the retina of the intended recipient of the message. [ 57 ]
Theoretical and mathematical evidence of an underlying quantum structure in human color perception has been presented by Michel Berthier and Edoardo Provenzi in a series of scientific articles. [ 58 ] [ 59 ] Notably, in their quantum formalism, the chromatic opposition phenomena proposed by Hering emerge naturally. Uncertainty principles for the perception of opposition have been predicted within this framework, which has so far demonstrated concrete applications in the removal of color cast in natural images caused by the presence of a non-neutral illuminant. [ 60 ]
Photosynthesis refers to the biological process that photosynthetic cells use to synthesize organic compounds from inorganic starting materials using sunlight. [ 61 ] What has been primarily implicated as exhibiting non-trivial quantum behaviors is the light reaction stage of photosynthesis. In this stage, photons are absorbed by the membrane-bound photosystems . Photosystems contain two major domains, the light-harvesting complex (antennae) and the reaction center . These antennae vary among organisms. For example, bacteria use circular aggregates of chlorophyll pigments, while plants use membrane-embedded protein and chlorophyll complexes. [ 62 ] [ 63 ] Regardless, photons are first captured by the antennae and passed on to the reaction-center complex. Various pigment-protein complexes, such as the FMO complex in green sulfur bacteria, are responsible for transferring energy from antennae to reaction site. The photon-driven excitation of the reaction-center complex mediates the oxidation and the reduction of the primary electron acceptor, a component of the reaction-center complex. Much like the electron transport chain of the mitochondria, a linear series of oxidations and reductions drives proton (H+) pumping across the thylakoid membrane, the development of a proton motive force , and energetic coupling to the synthesis of ATP .
Previous understandings of electron-excitation transference (EET) from light-harvesting antennae to the reaction center have relied on the Förster theory of incoherent EET, postulating weak electron coupling between chromophores and incoherent hopping from one to another. This theory has largely been disproven by FT electron spectroscopy experiments that show electron absorption and transfer with an efficiency of above 99%, [ 64 ] which cannot be explained by classical mechanical models. Instead, as early as 1938, scientists theorized that quantum coherence was the mechanism for excitation-energy transfer. Indeed, the structure and nature of the photosystem places it in the quantum realm, with EET ranging from the femto- to nanosecond scale, covering sub-nanometer to nanometer distances. [ 65 ] The effects of quantum coherence on EET in photosynthesis are best understood through state and process coherence. State coherence refers to the extent of individual superpositions of ground and excited states for quantum entities, such as excitons . Process coherence, on the other hand, refers to the degree of coupling between multiple quantum entities and their evolution as either dominated by unitary or dissipative parts, which compete with one another. Both of these types of coherence are implicated in photosynthetic EET, where a exciton is coherently delocalized over several chromophores. [ 66 ] This delocalization allows for the system to simultaneously explore several energy paths and use constructive and destructive interference to guide the path of the exciton's wave packet. It is presumed that natural selection has favored the most efficient path to the reaction center. Experimentally, the interaction between the different frequency wave packets, made possible by long-lived coherence, will produce quantum beats . [ 67 ]
While quantum photosynthesis is still an emerging field, there have been many experimental results that support the quantum-coherence understanding of photosynthetic EET. A 2007 study claimed the identification of electronic quantum coherence [ 68 ] at −196 °C (77 K). Another theoretical study from 2010 [ which? ] provided evidence that quantum coherence lives as long as 300 femtoseconds at biologically relevant temperatures (4 °C or 277 K). In that same year, experiments conducted on photosynthetic cryptophyte algae using two-dimensional photon echo spectroscopy yielded further confirmation for long-term quantum coherence. [ 69 ] These studies suggest that, through evolution, nature has developed a way of protecting quantum coherence to enhance the efficiency of photosynthesis. However, critical follow-up studies question the interpretation of these results. Single-molecule spectroscopy now shows the quantum characteristics of photosynthesis without the interference of static disorder, and some studies use this method to assign reported signatures of electronic quantum coherence to nuclear dynamics occurring in chromophores. [ 70 ] [ 71 ] [ 72 ] [ 73 ] [ 74 ] [ 75 ] [ 76 ] A number of proposals emerged to explain unexpectedly long coherence. According to one proposal, if each site within the complex feels its own environmental noise, the electron will not remain in any local minimum due to both quantum coherence and its thermal environment, but proceed to the reaction site via quantum walks . [ 77 ] [ 78 ] [ 79 ] Another proposal is that the rate of quantum coherence and electron tunneling create an energy sink that moves the electron to the reaction site quickly. [ 80 ] Other work suggested that geometric symmetries in the complex may favor efficient energy transfer to the reaction center, mirroring perfect state transfer in quantum networks. [ 81 ] Furthermore, experiments with artificial dye molecules cast doubts on the interpretation that quantum effects last any longer than one hundred femtoseconds. [ 82 ]
In 2017, the first control experiment with the original FMO protein under ambient conditions confirmed that electronic quantum effects are washed out within 60 femtoseconds, while the overall exciton transfer takes a time on the order of a few picoseconds. [ 83 ] In 2020 a review based on a wide collection of control experiments and theory concluded that the proposed quantum effects as long lived electronic coherences in the FMO system does not hold. [ 84 ] Instead, research investigating transport dynamics suggests that interactions between electronic and vibrational modes of excitation in FMO complexes require a semi-classical, semi-quantum explanation for the transfer of exciton energy. In other words, while quantum coherence dominates in the short-term, a classical description is most accurate to describe long-term behavior of the excitons. [ 85 ] [ 86 ]
Another process in photosynthesis that has almost 100% efficiency is charge transfer , again suggesting that quantum mechanical phenomena are at play. [ 76 ] In 1966, a study on the photosynthetic bacterium Chromatium found that at temperatures below 100 K, cytochrome oxidation is temperature-independent, slow (on the order of milliseconds), and very low in activation energy . The authors, Don DeVault and Britton Chase, postulated that these characteristics of electron transfer are indicative of quantum tunneling , whereby electrons penetrate a potential barrier despite possessing less energy than is classically necessary.
Mitochondria have been demonstrated to utilize quantum tunneling in their function as the powerhouse of eukaryotic cells. Similar to the light reactions in the thylakoid , linearly-associated membrane-bound proteins comprising the electron transport chain (ETC) energetically link the reduction of O2 with the development of a proton motive gradient (H+) across the inner membrane of the mitochondria. This energy stored as a proton motive gradient is then coupled with the synthesis of ATP . It is significant that the mitochondrion conversion of biomass into chemical ATP achieves 60-70% thermodynamic efficiency, far superior to that of man-made engines . [ 87 ] This high degree of efficiency is largely attributed to the quantum tunnelling of electrons in the ETC and of protons in the proton motive gradient. Indeed, electron tunneling has already been demonstrated in certain elements of the ETC including NADH:ubiquinone oxidoreductase (Complex I) and CoQH2-cytochrome c reductase (Complex III). [ 88 ] [ 89 ]
In quantum mechanics, both electrons and protons are quantum entities that exhibit wave-particle duality , exhibiting both particle and wave-like properties depending on the method of experimental observation. [ 90 ] Quantum tunneling is a direct consequence of this wave-like nature of quantum entities that permits the passing-through of a potential energy barrier that would otherwise restrict the entity. [ 91 ] Moreover, it depends on the shape and size of a potential barrier relative to the incoming energy of a particle. [ 92 ] Because the incoming particle is defined by its wave function, its tunneling probability is dependent upon the potential barrier's shape in an exponential way. For example, if the barrier is relatively wide, the incoming particle's probability to tunnel will decrease. The potential barrier, in some sense, can come in the form of an actual biomaterial barrier. The inner mitochondria membrane which houses the various components of the ETC is on the order of 7.5 nm thick. [ 87 ] The inner membrane of a mitochondrion must be overcome to permit signals (in the form of electrons, protons, H + ) to transfer from the site of emittance (internal to the mitochondria) and the site of acceptance (i.e. the electron transport chain proteins). [ 93 ] In order to transfer particles, the membrane of the mitochondria must have the correct density of phospholipids to conduct a relevant charge distribution that attracts the particle in question. For instance, for a greater density of phospholipids, the membrane contributes to a greater conductance of protons. [ 93 ]
Alexander Davydov developed the quantum theory of molecular solitons in order to explain the transport of energy in protein α-helices in general and the physiology of muscle contraction in particular. [ 94 ] [ 95 ] He showed that the molecular solitons are able to preserve their shape through nonlinear interaction of amide I excitons and phonon deformations inside the lattice of hydrogen-bonded peptide groups . [ 96 ] [ 97 ] In 1979, Davydov published his complete textbook on quantum biology entitled "Biology and Quantum Mechanics" featuring quantum dynamics of proteins , cell membranes , bioenergetics , muscle contraction , and electron transport in biomolecules . [ 11 ] [ 12 ]
Magnetoreception is the ability of animals to navigate using the inclination of the magnetic field of the Earth. [ 99 ] A possible explanation for magnetoreception is the entangled radical pair mechanism . [ 100 ] [ 101 ] The radical-pair mechanism is well-established in spin chemistry , [ 102 ] [ 103 ] [ 104 ] and was speculated to apply to magnetoreception in 1978 by Schulten et al.. The ratio between singlet and triplet pairs is changed by the interaction of entangled electron pairs with the magnetic field of the Earth. [ 105 ] In 2000, cryptochrome was proposed as the "magnetic molecule" that could harbor magnetically sensitive radical-pairs. Cryptochrome, a flavoprotein found in the eyes of European robins and other animal species, is the only protein known to form photoinduced radical-pairs in animals. [ 99 ] When it interacts with light particles, cryptochrome goes through a redox reaction, which yields radical pairs both during the photo-reduction and the oxidation. The function of cryptochrome is diverse across species, however, the photoinduction of radical-pairs occurs by exposure to blue light, which excites an electron in a chromophore . [ 105 ] Magnetoreception is also possible in the dark, so the mechanism must rely more on the radical pairs generated during light-independent oxidation.
Experiments in the lab support the basic theory that radical-pair electrons can be significantly influenced by very weak magnetic fields, i.e., merely the direction of weak magnetic fields can affect radical-pair's reactivity and therefore can "catalyze" the formation of chemical products. Whether this mechanism applies to magnetoreception and/or quantum biology, that is, whether Earth's magnetic field "catalyzes" the formation of bio chemical products by the aid of radical-pairs, is not fully clear. Radical-pairs may need not be entangled, the key quantum feature of the radical-pair mechanism, to play a part in these processes. There are entangled and non-entangled radical-pairs, but disturbing only entangled radical-pairs is not possible with current technology. Researchers found evidence for the radical-pair mechanism of magnetoreception when European robins, cockroaches, and garden warblers, could no longer navigate when exposed to a radio frequency that obstructs magnetic fields [ 99 ] and radical-pair chemistry. Further evidence came from a comparison of Cryptochrome 4 (CRY4) from migrating and non-migrating birds. CRY4 from chicken and pigeon were found to be less sensitive to magnetic fields than those from the (migrating) European robin , suggesting evolutionary optimization of this protein as a sensor of magnetic fields. [ 106 ]
DNA acts as the instructions for making proteins throughout the body. It consists of 4 nucleotides: guanine, thymine, cytosine, and adenine. [ 107 ] The order of these nucleotides gives the "recipe" for the different proteins.
Whenever a cell reproduces, it must copy these strands of DNA. However, sometime throughout the process of copying the strand of DNA a mutation, or an error in the DNA code, can occur. A theory for the reasoning behind DNA mutation is explained in the Lowdin DNA mutation model. [ 108 ] In this model, a nucleotide may spontaneously change its form through a process of quantum tunneling . [ 109 ] [ 110 ] Because of this, the changed nucleotide will lose its ability to pair with its original base pair and consequently change the structure and order of the DNA strand.
Exposure to ultraviolet light and other types of radiation can cause DNA mutation and damage. The radiation also can modify the bonds along the DNA strand in the pyrimidines and cause them to bond with themselves, creating a dimer. [ 111 ]
In many prokaryotes and plants, these bonds are repaired by a DNA-repair-enzyme photolyase. As its prefix implies, photolyase is reliant on light in order to repair the strand. Photolyase works with its cofactor FADH , flavin adenine dinucleotide, while repairing the DNA. Photolyase is excited by visible light and transfers an electron to the cofactor FADH. FADH—now in the possession of an extra electron—transfers the electron to the dimer to break the bond and repair the DNA. The electron tunnels from the FADH to the dimer . Although the range of this tunneling is much larger than feasible in a vacuum, the tunneling in this scenario is said to be "superexchange-mediated tunneling," and is possible due to the protein's ability to boost the tunneling rates of the electron. [ 108 ]
Other quantum phenomena in biological systems include the conversion of chemical energy into motion [ 112 ] and brownian motors in many cellular processes. [ 113 ]
Alongside the multiple strands of scientific inquiry into quantum mechanics has come unconnected pseudoscientific interest; this caused scientists to approach quantum biology cautiously. [ 114 ]
Hypotheses such as orchestrated objective reduction which postulate a link between quantum mechanics and consciousness have drawn criticism from the scientific community with some claiming it to be pseudoscientific and "an excuse for quackery". [ 115 ] | https://en.wikipedia.org/wiki/Quantum_biology |
The quantum boomerang effect is a quantum mechanical phenomenon whereby wavepackets launched through disordered media return, on average, to their starting points, as a consequence of Anderson localization and the inherent symmetries of the system. At early times, the initial parity asymmetry of the nonzero momentum leads to asymmetric behavior: nonzero displacement of the wavepackets from their origin. At long times, inherent time-reversal symmetry and the confining effects of Anderson localization lead to correspondingly symmetric behavior: both zero final velocity and zero final displacement. [ 1 ]
In 1958, Philip W. Anderson introduced the eponymous model of disordered lattices which exhibits localization, the confinement of the electrons' probability distributions within some small volume. [ 2 ] In other words, if a wavepacket were dropped into a disordered medium, it would spread out initially but then approach some maximum range. On the macroscopic scale, the transport properties of the lattice are reduced as a result of localization, turning what might have been a conductor into an insulator . Modern condensed matter models continue to study disorder as an important feature of real, imperfect materials. [ 3 ]
In 2019, theorists considered the behavior of a wavepacket not merely dropped, but actively launched through a disordered medium with some initial nonzero momentum , predicting that the wavepacket's center of mass would asymptotically return to the origin at long times — the quantum boomerang effect. [ 1 ] Shortly after, quantum simulation experiments in cold atom settings confirmed this prediction [ 4 ] [ 5 ] [ 6 ] by simulating the quantum kicked rotor , a model that maps to the Anderson model of disordered lattices. [ 7 ]
Consider a wavepacket Ψ ( x , t ) ∝ exp [ − x 2 / ( 2 σ ) 2 + i k 0 x ] {\displaystyle \Psi (x,t)\propto \exp \left[-x^{2}/(2\sigma )^{2}+ik_{0}x\right]} with initial momentum ℏ k 0 {\displaystyle \hbar k_{0}} which evolves in the general Hamiltonian of a Gaussian, uncorrelated, disordered medium:
where V ( x ) ¯ = 0 {\displaystyle {\overline {V(x)}}=0} and V ( x ) V ( x ′ ) ¯ = γ δ ( x − x ′ ) {\displaystyle {\overline {V(x)V(x')}}=\gamma \delta (x-x')} , and the overbar notation indicates an average over all possible realizations of the disorder.
The classical Boltzmann equation predicts that this wavepacket should slow down and localize at some new point — namely, the terminus of its mean free path. However, when accounting for the quantum mechanical effects of localization and time-reversal symmetry (or some other unitary or antiunitary symmetry [ 8 ] ), the probability density distribution | Ψ 2 | {\displaystyle |\Psi ^{2}|} exhibits off-diagonal, oscillatory elements in its eigenbasis expansion that decay at long times, leaving behind only diagonal elements independent of the sign of the initial momentum. Since the direction of the launch does not matter at long times, the wavepacket must return to the origin. [ 1 ]
The same destructive interference argument used to justify Anderson localization applies to the quantum boomerang. The Ehrenfest theorem states that the variance ( i.e. the spread) of the wavepacket evolves thus:
where the use of the Wigner function allows the final approximation of the particle distribution into two populations n ± {\displaystyle n_{\pm }} of positive and negative velocities, with centers of mass denoted
A path contributing to ⟨ x ^ ⟩ − {\displaystyle \langle {\hat {x}}\rangle _{-}} at some time must have negative momentum − ℏ k 0 {\displaystyle -\hbar k_{0}} by definition; since every part of the wavepacket originated at the same positive momentum ℏ k 0 {\displaystyle \hbar k_{0}} behavior, this path from the origin to x {\displaystyle x} and from initial ℏ k 0 {\displaystyle \hbar k_{0}} momentum to final − ℏ k 0 {\displaystyle -\hbar k_{0}} momentum can be time-reversed and translated to create another path from x {\displaystyle x} back to the origin with the same initial and final momenta. This second, time-reversed path is equally weighted in the calculation of n − ( x , t ) {\displaystyle n_{-}(x,t)} and ultimately results in ⟨ x ^ ⟩ − = 0 {\displaystyle \langle {\hat {x}}\rangle _{-}=0} . The same logic does not apply to ⟨ x ^ ⟩ + {\displaystyle \langle {\hat {x}}\rangle _{+}} because there is no initial population in the momentum state − ℏ k 0 {\displaystyle -\hbar k_{0}} . Thus, the wavepacket variance only has the first term:
This yields long-time behavior
where ℓ {\displaystyle \ell } and τ {\displaystyle \tau } are the scattering mean free path and scattering mean free time , respectively. The exact form of the boomerang can be approximated using the diagonal Padé approximants R [ n / n ] {\displaystyle R_{[n/n]}} extracted from a series expansion derived with the Berezinskii diagrammatic technique. [ 1 ] | https://en.wikipedia.org/wiki/Quantum_boomerang_effect |
Quantum calculus , sometimes called calculus without limits , is equivalent to traditional infinitesimal calculus without the notion of limits . The two types of calculus in quantum calculus are q -calculus and h -calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit , the original object is returned. In q -calculus, the limit as q tends to 1 is taken of the q -analog . Likewise, in h -calculus, the limit as h tends to 0 is taken of the h -analog. The parameters q {\displaystyle q} and h {\displaystyle h} can be related by the formula q = e h {\displaystyle q=e^{h}} .
The q -differential and h -differential are defined as:
and
respectively. The q -derivative and h -derivative are then defined as
and
respectively. By taking the limit as q → 1 {\displaystyle q\rightarrow 1} of the q -derivative or as h → 0 {\displaystyle h\rightarrow 0} of the h -derivative, one can obtain the derivative :
A function F ( x ) is a q-antiderivative of f ( x ) if D q F ( x ) = f ( x ). The q-antiderivative (or q-integral) is denoted by ∫ f ( x ) d q x {\textstyle \int f(x)\,d_{q}x} and an expression for F ( x ) can be found from: ∫ f ( x ) d q x = ( 1 − q ) ∑ j = 0 ∞ x q j f ( x q j ) {\textstyle \int f(x)\,d_{q}x=(1-q)\sum _{j=0}^{\infty }xq^{j}f(xq^{j})} , which is called the Jackson integral of f ( x ). For 0 < q < 1 , the series converges to a function F ( x ) on an interval (0, A ] if | f ( x ) x α | is bounded on the interval (0, A ] for some 0 ≤ α < 1 .
The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q j. .The jump at the point q j is q j . Calling this step function g q ( t ) gives dg q ( t ) = d q t . [ 1 ]
A function F ( x ) is an h-antiderivative of f ( x ) if D h F ( x ) = f ( x ). The h-integral is denoted by ∫ f ( x ) d h x {\textstyle \int f(x)\,d_{h}x} . If a and b differ by an integer multiple of h then the definite integral ∫ a b f ( x ) d h x {\textstyle \int _{a}^{b}f(x)\,d_{h}x} is given by a Riemann sum of f ( x ) on the interval [ a , b ] , partitioned into sub-intervals of equal width h . The motivation of h-integral comes from the Riemann sum of f(x). Following the idea of the motivation of classical integrals, some of the properties of classical integrals hold in h-integral. This notion has broad applications in numerical analysis , and especially finite difference calculus .
In infinitesimal calculus, the derivative of the function x n {\displaystyle x^{n}} is n x n − 1 {\displaystyle nx^{n-1}} (for some positive integer n {\displaystyle n} ). The corresponding expressions in q -calculus and h -calculus are:
where [ n ] q {\displaystyle [n]_{q}} is the q -bracket
and
respectively. The expression [ n ] q x n − 1 {\displaystyle [n]_{q}x^{n-1}} is then the q -analog and ∑ k = 1 n ( n k ) x n − k h k − 1 {\textstyle \sum _{k=1}^{n}{{\binom {n}{k}}x^{n-k}h^{k-1}}} is the h -analog of the power rule for positive integral powers. The q-Taylor expansion allows for the definition of q -analogs of all of the usual functions, such as the sine function, whose q -derivative is the q -analog of cosine .
The h -calculus is the calculus of finite differences , which was studied by George Boole and others, and has proven useful in combinatorics and fluid mechanics . In a sense, q -calculus dates back to Leonhard Euler and Carl Gustav Jacobi , but has only recently begun to find usefulness in quantum mechanics , given its intimate connection with commutativity relations and Lie algebras , specifically quantum groups . | https://en.wikipedia.org/wiki/Quantum_calculus |
In the theory of quantum communication , the quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to a receiver. It is also equal to the highest rate at which entanglement can be generated over the channel, and forward classical communication cannot improve it. The quantum capacity theorem is important for the theory of quantum error correction , and more broadly for the theory of quantum computation . The theorem giving a lower bound on the quantum capacity of any channel is colloquially known as the LSD theorem, after the authors Lloyd , [ 1 ] Shor , [ 2 ] and Devetak [ 3 ] who proved it with increasing standards of rigor. [ 4 ]
The LSD theorem states that the coherent information of a quantum channel is an achievable rate for reliable quantum communication. For a Pauli channel , the coherent information has a simple form [ citation needed ] and the proof that it is achievable is particularly simple as well. We [ who? ] prove the theorem for this special case by exploiting random stabilizer codes and correcting only the likely errors that the channel produces.
Theorem (hashing bound). There exists a stabilizer quantum error-correcting code that achieves the hashing limit R = 1 − H ( p ) {\displaystyle R=1-H\left(\mathbf {p} \right)} for a Pauli channel of the following form: ρ ↦ p I ρ + p X X ρ X + p Y Y ρ Y + p Z Z ρ Z , {\displaystyle \rho \mapsto p_{I}\rho +p_{X}X\rho X+p_{Y}Y\rho Y+p_{Z}Z\rho Z,} where p = ( p I , p X , p Y , p Z ) {\displaystyle \mathbf {p} =\left(p_{I},p_{X},p_{Y},p_{Z}\right)} and H ( p ) {\displaystyle H\left(\mathbf {p} \right)} is the entropy of this probability vector.
Proof . Consider correcting only the typical errors. That is, consider defining the typical set of errors as follows: T δ p n ≡ { a n : | − 1 n log 2 ( Pr { E a n } ) − H ( p ) | ≤ δ } , {\displaystyle T_{\delta }^{\mathbf {p} ^{n}}\equiv \left\{a^{n}:\left\vert -{\frac {1}{n}}\log _{2}\left(\Pr \left\{E_{a^{n}}\right\}\right)-H\left(\mathbf {p} \right)\right\vert \leq \delta \right\},} where a n {\displaystyle a^{n}} is some sequence consisting of the letters { I , X , Y , Z } {\displaystyle \left\{I,X,Y,Z\right\}} and Pr { E a n } {\displaystyle \Pr \left\{E_{a^{n}}\right\}} is the probability that an IID Pauli channel issues some tensor-product error E a n ≡ E a 1 ⊗ ⋯ ⊗ E a n {\displaystyle E_{a^{n}}\equiv E_{a_{1}}\otimes \cdots \otimes E_{a_{n}}} . This typical set consists of the likely errors in the sense that ∑ a n ∈ T δ p n Pr { E a n } ≥ 1 − ϵ , {\displaystyle \sum _{a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}}\Pr \left\{E_{a^{n}}\right\}\geq 1-\epsilon ,} for all ϵ > 0 {\displaystyle \epsilon >0} and sufficiently large n {\displaystyle n} . The error-correcting
conditions [ 5 ] for a stabilizer code S {\displaystyle {\mathcal {S}}} in this case are that { E a n : a n ∈ T δ p n } {\displaystyle \{E_{a^{n}}:a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}\}} is a correctable set of errors if
E a n † E b n ∉ N ( S ) ∖ S , {\displaystyle E_{a^{n}}^{\dagger }E_{b^{n}}\notin N\left({\mathcal {S}}\right)\backslash {\mathcal {S}},} for all error pairs E a n {\displaystyle E_{a^{n}}} and E b n {\displaystyle E_{b^{n}}} such that a n , b n ∈ T δ p n {\displaystyle a^{n},b^{n}\in T_{\delta }^{\mathbf {p} ^{n}}} where N ( S ) {\displaystyle N({\mathcal {S}})} is the normalizer of S {\displaystyle {\mathcal {S}}} . Also, we consider the expectation of the error probability under a random choice of a stabilizer code.
Proceed as follows: E S { p e } = E S { ∑ a n Pr { E a n } I ( E a n is uncorrectable under S ) } ≤ E S { ∑ a n ∈ T δ p n Pr { E a n } I ( E a n is uncorrectable under S ) } + ϵ = ∑ a n ∈ T δ p n Pr { E a n } E S { I ( E a n is uncorrectable under S ) } + ϵ = ∑ a n ∈ T δ p n Pr { E a n } Pr S { E a n is uncorrectable under S } + ϵ . {\displaystyle {\begin{aligned}\mathbb {E} _{\mathcal {S}}\left\{p_{e}\right\}&=\mathbb {E} _{\mathcal {S}}\left\{\sum _{a^{n}}\Pr \left\{E_{a^{n}}\right\}{\mathcal {I}}\left(E_{a^{n}}{\text{ is uncorrectable under }}{\mathcal {S}}\right)\right\}\\&\leq \mathbb {E} _{\mathcal {S}}\left\{\sum _{a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}}\Pr \left\{E_{a^{n}}\right\}{\mathcal {I}}\left(E_{a^{n}}{\text{ is uncorrectable under }}{\mathcal {S}}\right)\right\}+\epsilon \\&=\sum _{a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}}\Pr \left\{E_{a^{n}}\right\}\mathbb {E} _{\mathcal {S}}\left\{{\mathcal {I}}\left(E_{a^{n}}{\text{ is uncorrectable under }}{\mathcal {S}}\right)\right\}+\epsilon \\&=\sum _{a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}}\Pr \left\{E_{a^{n}}\right\}\Pr _{\mathcal {S}}\left\{E_{a^{n}}{\text{ is uncorrectable under }}{\mathcal {S}}\right\}+\epsilon .\end{aligned}}} The first equality follows by definition— I {\displaystyle {\mathcal {I}}} is an indicator function equal to one if E a n {\displaystyle E_{a^{n}}} is uncorrectable under S {\displaystyle {\mathcal {S}}} and equal to zero otherwise. The first inequality follows, since we correct only the typical errors because the atypical error set has negligible probability mass. The second equality follows by exchanging the expectation and the sum. The third equality follows because the expectation of an indicator function is the probability that the event it selects occurs.
Continuing, we have: = ∑ a n ∈ T δ p n Pr { E a n } Pr S { ∃ E b n : b n ∈ T δ p n , b n ≠ a n , E a n † E b n ∈ N ( S ) ∖ S } {\displaystyle =\sum _{a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}}\Pr \left\{E_{a^{n}}\right\}\Pr _{\mathcal {S}}\left\{\exists E_{b^{n}}:b^{n}\in T_{\delta }^{\mathbf {p} ^{n}},\ b^{n}\neq a^{n},\ E_{a^{n}}^{\dagger }E_{b^{n}}\in N\left({\mathcal {S}}\right)\backslash {\mathcal {S}}\right\}}
The first equality follows from the error-correcting conditions for a quantum stabilizer code, where N ( S ) {\displaystyle N\left({\mathcal {S}}\right)} is the normalizer of S {\displaystyle {\mathcal {S}}} . The first inequality follows by ignoring any potential degeneracy in the code—we consider an error uncorrectable if it lies in the normalizer N ( S ) {\displaystyle N\left({\mathcal {S}}\right)} and the probability can only be larger because N ( S ) ∖ S ∈ N ( S ) {\displaystyle N\left({\mathcal {S}}\right)\backslash {\mathcal {S}}\in N\left({\mathcal {S}}\right)} . The second equality follows by realizing that the probabilities for the existence criterion and the union of events are equivalent. The second inequality follows by applying the union bound. The third inequality follows from the fact that the probability for a fixed operator E a n † E b n {\displaystyle E_{a^{n}}^{\dagger }E_{b^{n}}} not equal to the identity commuting with
the stabilizer operators of a random stabilizer can be upper bounded as follows: Pr S { E a n † E b n ∈ N ( S ) } = 2 n + k − 1 2 2 n − 1 ≤ 2 − ( n − k ) . {\displaystyle \Pr _{\mathcal {S}}\left\{E_{a^{n}}^{\dagger }E_{b^{n}}\in N\left({\mathcal {S}}\right)\right\}={\frac {2^{n+k}-1}{2^{2n}-1}}\leq 2^{-\left(n-k\right)}.} The reasoning here is that the random choice of a stabilizer code is equivalent to
fixing operators Z 1 {\displaystyle Z_{1}} , ..., Z n − k {\displaystyle Z_{n-k}} and performing a uniformly random
Clifford unitary. The probability that a fixed operator commutes with Z ¯ 1 {\displaystyle {\overline {Z}}_{1}} , ..., Z ¯ n − k {\displaystyle {\overline {Z}}_{n-k}} is then just the number of
non-identity operators in the normalizer ( 2 n + k − 1 {\displaystyle 2^{n+k}-1} ) divided by the total number of non-identity operators ( 2 2 n − 1 {\displaystyle 2^{2n}-1} ). After applying the above bound, we then exploit the following typicality bounds: ∀ a n ∈ T δ p n : Pr { E a n } ≤ 2 − n [ H ( p ) + δ ] , {\displaystyle \forall a^{n}\in T_{\delta }^{\mathbf {p} ^{n}}:\Pr \left\{E_{a^{n}}\right\}\leq 2^{-n\left[H\left(\mathbf {p} \right)+\delta \right]},} | T δ p n | ≤ 2 n [ H ( p ) + δ ] . {\displaystyle \left\vert T_{\delta }^{\mathbf {p} ^{n}}\right\vert \leq 2^{n\left[H\left(\mathbf {p} \right)+\delta \right]}.} We conclude that as long as the rate k / n = 1 − H ( p ) − 4 δ {\displaystyle k/n=1-H\left(\mathbf {p} \right)-4\delta } , the expectation of the error probability becomes arbitrarily small, so that there exists at least one choice of a stabilizer code with the same bound on the error probability. | https://en.wikipedia.org/wiki/Quantum_capacity |
In quantum mechanics , a quantum carpet [ 1 ] is a regular art -like pattern drawn by the wave function evolution or the probability density in the space of the Cartesian product of the quantum particle position coordinate and time or in spacetime resembling carpet art. It is the result of self- interference of the wave function during its interaction with reflecting boundaries. For example, in the infinite potential well , after the spread of the initially localized Gaussian wave packet in the center of the well, various pieces of the wave function start to overlap and interfere with each other after reflection from the boundaries. The geometry of a quantum carpet is mainly determined by the quantum fractional revivals .
Quantum carpets demonstrate many principles of quantum mechanics, including wave-particle duality , quantum revivals , and decoherence . Thus, they illustrate certain aspects of theoretical physics.
In 1995, Michael Berry created the first quantum carpet, which described the momentum of an excited atom. Today, physicists use quantum carpets to demonstrate complex theoretical principles. [ 2 ] [ 3 ]
Quantum carpets demonstrate wave-particle duality by showing interference within wave packets.
Wave particle duality is difficult to comprehend. However, quantum carpets provide an opportunity to visualize this property. Consider the graph of the probability distribution of an excited electron in a confined space (particle in a box), where brightness of color corresponds to momentum. Lines of dull color (ghost terms or canals) appear across the quantum carpet. In these canals, the momentum of the electron is very small. Destructive interference , when the trough of a wave overlaps with the crest of another wave, causes these ghost terms. In contrast, some areas of the graph display bright color. Constructive interference , when the crests of two waves overlap to form a larger wave, causes these bright colors. Thus, quantum carpets provide visual evidence of interference within electrons and other wave packets. Interference is a property of waves, not particles, so interference within these wave packets prove that they have properties of waves in addition to properties of particles. Therefore, quantum carpets display wave particle duality. [ 4 ]
Quantum carpets demonstrate quantum revivals by showing the periodic expansions and contractions of wave packets.
When the momentum of a wave packet is graphed on a quantum carpet, it displays an intricate pattern. When the temporal evolution of this wave packet is graphed on quantum carpets, the wave packet expands, and the initial pattern is lost. However, after a certain period of time, the waveform contracts and returns to its original state, and the initial pattern is restored. [ 5 ] This continues to occur with periodic regularity. Quantum revivals, the periodic expansion and contraction of wave packets, are responsible for the restoration of the pattern. [ 6 ] Although quantum revivals are mathematically complex, they are simple and easy to visualize on quantum carpets, as patterns expanding and reforming. Thus, quantum carpets provide clear visual evidence of quantum revivals.
Quantum carpets demonstrate decoherence by showing a loss of coherence over time.
When the temporal evolution of an electron, photon, or atom is graphed on a quantum carpet, there is initially a distinct pattern. This distinct pattern shows coherence. That is to say, the wave can be split in two pieces and recombined to form a new wave. [ 7 ] However, this pattern fades with time, and eventually, devolves into nothing. When the pattern fades, coherence is lost, and it is impossible to split the wave in two and recombine it. This loss of coherence is called decoherence. [ 8 ] A set of complex mathematical equations model decoherence. However, a simple loss of pattern shows decoherence in quantum carpets. Thus, quantum carpets are a tool to visualize and simplify decoherence.
While performing an experiment on optics, English physicist Henry Fox Talbot inadvertently discovered the key to quantum carpets. In this experiment, a wave struck a diffraction grating , and Talbot noticed that the patterns of grating repeated themselves with periodic regularity. This phenomenon became known as the Talbot Effect . The bands of light that Talbot discovered were never graphed on an axis, and thus, he never created a true quantum carpet. [ 9 ] However, the bands of light were similar to the images on a quantum carpet. Centuries later, physicists graphed the Talbot effect, creating the first quantum carpet. Since then, scientists have turned to quantum carpets as visual evidence for quantum theory. [ 2 ]
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_carpet |
Quantum chaos is a branch of physics focused on how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos ?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of the Planck constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics? [ 1 ] [ 2 ]
In seeking to address the basic question of quantum chaos, several approaches have been employed:
During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the three-body problem in celestial mechanics ), but not well understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibit chaos.
Questions related to the correspondence principle arise in many different branches of physics, ranging from nuclear to atomic , molecular and solid-state physics , and even to acoustics , microwaves and optics . However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics. [ 5 ]
Important observations often associated with classically chaotic quantum systems are spectral level repulsion , dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in scattering ). In the semiclassical approach of quantum chaos, phenomena are identified in spectroscopy by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the time evolution of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude distributions.
Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. Dyson series ). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.
For conservative systems, the goal of quantum mechanics in non-perturbative regimes is to find
the eigenvalues and eigenvectors of a Hamiltonian of the form
where H s {\displaystyle H_{s}} is separable in some coordinate system, H n s {\displaystyle H_{ns}} is non-separable in the coordinate system in which H s {\displaystyle H_{s}} is separated, and ϵ {\displaystyle \epsilon } is a parameter which cannot be considered small. Physicists have historically approached problems of this nature by trying to find the coordinate system in which the non-separable Hamiltonian is smallest and then treating the non-separable Hamiltonian as a perturbation.
Finding constants of motion so that this separation can be performed can be a difficult (sometimes impossible) analytical task. Solving the classical problem can give valuable insight into solving the quantum problem. If there are regular classical solutions of
the same Hamiltonian, then there are (at least) approximate constants of motion, and by solving the classical problem, we gain clues how to find them.
Other approaches have been developed in recent years. One is to express the Hamiltonian in
different coordinate systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region. Wavefunctions are obtained in these regions, and eigenvalues are obtained by matching boundary conditions.
Another approach is numerical matrix diagonalization. If the Hamiltonian matrix is computed in any complete basis, eigenvalues and eigenvectors are obtained by diagonalizing the matrix. However, all complete basis sets are infinite, and we need to truncate the basis and still obtain accurate results. These techniques boil down to choosing a truncated basis from which accurate wavefunctions can be constructed. The computational time required to diagonalize a matrix scales as N 3 {\displaystyle N^{3}} , where N {\displaystyle N} is the dimension of the matrix, so it is important to choose the smallest basis possible from which the relevant wavefunctions can be constructed. It is also convenient to choose a basis in which the matrix is sparse and/or the matrix elements are given by simple algebraic expressions because computing matrix elements can also be a computational burden.
A given Hamiltonian shares the same constants of motion for both classical and quantum dynamics. Quantum systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity conservation from reflection symmetry). However, if we merely find quantum solutions of a Hamiltonian which is not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of answering the question of quantum chaos.
Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Random matrix theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.
A number of statistical measures are available for quantifying spectral features in a simple way. It is of great interest whether or not there are universal statistical behaviors of classically chaotic systems. The statistical tests mentioned here are universal, at least to systems with few degrees of freedom ( Berry and Tabor [ 6 ] have put forward strong arguments for a Poisson distribution in the case of regular motion and Heusler et al. [ 7 ] present a semiclassical explanation of the so-called Bohigas–Giannoni–Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics). The nearest-neighbor distribution (NND) of energy levels is relatively simple to interpret and it has been widely used to describe quantum chaos.
Qualitative observations of level repulsions can be quantified and related to the classical dynamics using the NND, which is believed to be an important signature of classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a Poisson distribution of energy levels:
In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a number of chaotic systems have been shown to be in good agreement with the predictions of the Gaussian orthogonal ensemble (GOE) of random matrices, and it has been suggested that this phenomenon is generic for all chaotic systems with this symmetry. If the normalized spacing between two energy levels is s {\displaystyle s} , the normalized distribution of spacings is well approximated by
Many Hamiltonian systems which are classically integrable (non-chaotic) have been found to have quantum solutions that yield nearest neighbor distributions which follow the Poisson distributions. Similarly, many systems which exhibit classical chaos have been found with quantum solutions yielding a Wigner-Dyson distribution , thus supporting the ideas above. One notable exception is diamagnetic lithium which, though exhibiting classical chaos, demonstrates Wigner (chaotic) statistics for the even-parity energy levels and nearly Poisson (regular) statistics for the odd-parity energy level distribution. [ 8 ]
Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. In contrast to the Einstein–Brillouin–Keller method of action quantization, which applies only to integrable or near-integrable systems and computes individual eigenvalues from each trajectory, periodic-orbit theory is applicable to both integrable and non-integrable systems and asserts that each periodic orbit produces a sinusoidal fluctuation in the density of states.
The principal result of this development is an expression for the density of states which is the trace of the semiclassical Green's function and is given by the Gutzwiller trace formula:
Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a Berry phase -like term stemming from spin or other internal degrees of freedom. [ 9 ] The index k {\displaystyle k} distinguishes the primitive periodic orbits : the shortest period orbits of a given set of initial conditions. T k {\displaystyle T_{k}} is the period of the primitive periodic orbit and S k {\displaystyle S_{k}} is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action n S k {\displaystyle nS_{k}} and a period which is an integral multiple n {\displaystyle n} of the primitive period. Hence, every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the intermediate sum over the indices n {\displaystyle n} . α n k {\displaystyle \alpha _{nk}} is the orbit's Maslov index .
The amplitude factor, 1 / sinh ( χ n k / 2 ) {\displaystyle 1/\sinh {(\chi _{nk}/2)}} , represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. The quantity χ n k {\displaystyle \chi _{nk}} characterizes the instability of the orbit. A stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. For stable orbits, sinh ( χ n k / 2 ) {\displaystyle \sinh {(\chi _{nk}/2)}} becomes sin ( χ n k / 2 ) {\displaystyle \sin {(\chi _{nk}/2)}} , where χ n k {\displaystyle \chi _{nk}} is the winding number of the periodic orbit. χ n k = 2 π m {\displaystyle \chi _{nk}=2\pi m} , where m {\displaystyle m} is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because sin ( χ n k / 2 ) = 0 {\displaystyle \sin {(\chi _{nk}/2)}=0} at a classical bifurcation . This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo- absorption spectrum .
Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action. 2) There are an infinite number of periodic orbits, and the convergence properties of periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to roundoff errors and details of the numerical integration.
Gutzwiller applied the trace formula to approach the anisotropic Kepler problem (a single particle in a 1 / r {\displaystyle 1/r} potential with an anisotropic mass tensor ) semiclassically. He found agreement with quantum computations for low lying (up to n = 6 {\displaystyle n=6} ) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies.
The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states (energy levels), one can use standard quantum mechanical perturbation theory to compute eigenvalues (energy levels) and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform.
Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books.
Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and J. Shaw. It is similar to
periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density (observable photo-absorption spectrum) from a specified initial state whereas periodic-orbit theory yields the density of states.
Only orbits that begin and end at the nucleus are important in closed-orbit theory. Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state. For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time.
According to closed-orbit theory, the average oscillator strength density at constant ϵ {\displaystyle \epsilon } is given by a smooth background plus an oscillatory sum of the form
ϕ n k {\displaystyle \phi _{\it {nk}}} is a phase that depends on the Maslov index and other details of the orbits. D n k i {\displaystyle D_{\it {nk}}^{i}} is the recurrence amplitude of a closed orbit for a given initial state (labeled i {\displaystyle i} ). It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave. For scaling systems such as Rydberg atoms in strong fields, the Fourier transform of an oscillator strength spectrum computed at fixed ϵ {\displaystyle \epsilon } as a function of w {\displaystyle w} is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to D n k i {\displaystyle D_{\it {nk}}^{i}} .
Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen, hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the H − {\displaystyle H^{-}} ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.
For the case of one-dimensional system with the boundary condition y ( 0 ) = 0 {\displaystyle y(0)=0} the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by d 1 / 2 d x 1 / 2 V − 1 ( x ) = 2 π d N ( x ) d x {\displaystyle {\frac {d^{1/2}}{dx^{1/2}}}V^{-1}(x)=2{\sqrt {\pi }}{\frac {dN(x)}{dx}}} here d N ( x ) d x {\displaystyle {\frac {dN(x)}{dx}}} is the density of states and V(x) is the classical potential of the particle, the half derivative of the inverse of the potential is related to the density of states as in the Wu–Sprung potential .
One open question remains understanding quantum chaos in systems that have finite-dimensional local Hilbert spaces for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such quantum many-body systems . [ 10 ] [ 11 ]
The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions of various chaotic Hamiltonian. For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation ( Quantum ergodicity ). However, a quantum eigenstate of a classically chaotic system can be scarred: [ 12 ] the probability density of the eigenstate is enhanced in the neighborhood of a periodic orbit, above the classical, statistically expected density along the orbit ( scars ). In particular, scars are both a striking visual example of classical-quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos. For example, this is evident in the perturbation-induced quantum scarring: [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] More specifically, in quantum dots perturbed by local potential bumps (impurities), some of the eigenstates are strongly scarred along periodic orbits of unperturbed classical counterpart.
Further studies concern the parametric ( R {\displaystyle R} ) dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the standard map and the kicked rotator are considered to be prototype problems.
Works are also focused in the study of driven chaotic systems, [ 18 ] where the Hamiltonian H ( x , p ; R ( t ) ) {\displaystyle H(x,p;R(t))} is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting many-body quantum systems far from semi-classical regimes as well as a large effort in quantum chaotic scattering. [ 19 ]
In 1977, Berry and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface , the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable . [ 20 ] [ 21 ] [ 22 ] | https://en.wikipedia.org/wiki/Quantum_chaos |
Quantum chemistry , also called molecular quantum mechanics , is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules , materials , and solutions at the atomic level. [ 1 ] These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed wave functions as well as to observable properties such as structures, spectra, and thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics .
Chemists rely heavily on spectroscopy through which information regarding the quantization of energy on a molecular scale can be obtained. Common methods are infra-red (IR) spectroscopy , nuclear magnetic resonance (NMR) spectroscopy , and scanning probe microscopy . Quantum chemistry may be applied to the prediction and verification of spectroscopic data as well as other experimental data.
Many quantum chemistry studies are focused on the electronic ground state and excited states of individual atoms and molecules as well as the study of reaction pathways and transition states that occur during chemical reactions . Spectroscopic properties may also be predicted. Typically, such studies assume the electronic wave function is adiabatically parameterized by the nuclear positions (i.e., the Born–Oppenheimer approximation ). A wide variety of approaches are used, including semi-empirical methods, density functional theory , Hartree–Fock calculations, quantum Monte Carlo methods, and coupled cluster methods.
Understanding electronic structure and molecular dynamics through the development of computational solutions to the Schrödinger equation is a central goal of quantum chemistry. Progress in the field depends on overcoming several challenges, including the need to increase the accuracy of the results for small molecular systems, and to also increase the size of large molecules that can be realistically subjected to computation, which is limited by scaling considerations — the computation time increases as a power of the number of atoms.
Some view the birth of quantum chemistry as starting with the discovery of the Schrödinger equation and its application to the hydrogen atom. However, a 1927 article of Walter Heitler (1904–1981) and Fritz London is often recognized as the first milestone in the history of quantum chemistry. [ 2 ] This was the first application of quantum mechanics to the diatomic hydrogen molecule , and thus to the phenomenon of the chemical bond. [ 3 ] However, prior to this a critical conceptual framework was provided by Gilbert N. Lewis in his 1916 paper The Atom and the Molecule , [ 4 ] wherein Lewis developed the first working model of valence electrons . Important contributions were also made by Yoshikatsu Sugiura [ 5 ] [ 6 ] and S.C. Wang. [ 7 ] A series of articles by Linus Pauling , written throughout the 1930s, integrated the work of Heitler, London, Sugiura, Wang, Lewis, and John C. Slater on the concept of valence and its quantum-mechanical basis into a new theoretical framework. [ 8 ] Many chemists were introduced to the field of quantum chemistry by Pauling's 1939 text The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry , wherein he summarized this work (referred to widely now as valence bond theory ) and explained quantum mechanics in a way which could be followed by chemists. [ 9 ] The text soon became a standard text at many universities. [ 10 ] In 1937, Hans Hellmann appears to have been the first to publish a book on quantum chemistry, in the Russian [ 11 ] and German languages. [ 12 ]
In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding. In addition to the investigators mentioned above, important progress and critical contributions were made in the early years of this field by Irving Langmuir , Robert S. Mulliken , Max Born , J. Robert Oppenheimer , Hans Hellmann , Maria Goeppert Mayer , Erich Hückel , Douglas Hartree , John Lennard-Jones , and Vladimir Fock .
The electronic structure of an atom or molecule is the quantum state of its electrons. [ 13 ] The first step in solving a quantum chemical problem is usually solving the Schrödinger equation (or Dirac equation in relativistic quantum chemistry ) with the electronic molecular Hamiltonian , usually making use of the Born–Oppenheimer (B–O) approximation. This is called determining the electronic structure of the molecule. [ 14 ] An exact solution for the non-relativistic Schrödinger equation can only be obtained for the hydrogen atom (though exact solutions for the bound state energies of the hydrogen molecular ion within the B-O approximation have been identified in terms of the generalized Lambert W function ). Since all other atomic and molecular systems involve the motions of three or more "particles", their Schrödinger equations cannot be solved analytically and so approximate and/or computational solutions must be sought. The process of seeking computational solutions to these problems is part of the field known as computational chemistry .
As mentioned above, Heitler and London's method was extended by Slater and Pauling to become the valence-bond (VB)
method. In this method, attention is primarily devoted to the pairwise interactions between atoms, and this method therefore correlates closely with classical chemists' drawings of bonds . It focuses on how the atomic orbitals of an atom combine to give individual chemical bonds when a molecule is formed, incorporating the two key concepts of orbital hybridization and resonance . [ 15 ]
An alternative approach to valence bond theory was developed in 1929 by Friedrich Hund and Robert S. Mulliken , in which electrons are described by mathematical functions delocalized over an entire molecule . The Hund–Mulliken approach or molecular orbital (MO) method is less intuitive to chemists, but has turned out capable of predicting spectroscopic properties better than the VB method. This approach is the conceptual basis of the Hartree–Fock method and further post-Hartree–Fock methods.
The Thomas–Fermi model was developed independently by Thomas and Fermi in 1927. This was the first attempt to describe many-electron systems on the basis of electronic density instead of wave functions , although it was not very successful in the treatment of entire molecules. The method did provide the basis for what is now known as density functional theory (DFT). Modern day DFT uses the Kohn–Sham method , where the density functional is split into four terms; the Kohn–Sham kinetic energy, an external potential, exchange and correlation energies. A large part of the focus on developing DFT is on improving the exchange and correlation terms. Though this method is less developed than post Hartree–Fock methods, its significantly lower computational requirements (scaling typically no worse than n 3 with respect to n basis functions, for the pure functionals) allow it to tackle larger polyatomic molecules and even macromolecules . This computational affordability and often comparable accuracy to MP2 and CCSD(T) (post-Hartree–Fock methods) has made it one of the most popular methods in computational chemistry .
A further step can consist of solving the Schrödinger equation with the total molecular Hamiltonian in order to study the motion of molecules. Direct solution of the Schrödinger equation is called quantum dynamics , whereas its solution within the semiclassical approximation is called semiclassical dynamics. Purely classical simulations of molecular motion are referred to as molecular dynamics (MD) . Another approach to dynamics is a hybrid framework known as mixed quantum-classical dynamics ; yet another hybrid framework uses the Feynman path integral formulation to add quantum corrections to molecular dynamics, which is called path integral molecular dynamics . Statistical approaches, using for example classical and quantum Monte Carlo methods , are also possible and are particularly useful for describing equilibrium distributions of states.
In adiabatic dynamics, interatomic interactions are represented by single scalar potentials called potential energy surfaces . This is the Born–Oppenheimer approximation introduced by Born and Oppenheimer in 1927. Pioneering applications of this in chemistry were performed by Rice and Ramsperger in 1927 and Kassel in 1928, and generalized into the RRKM theory in 1952 by Marcus who took the transition state theory developed by Eyring in 1935 into account. These methods enable simple estimates of unimolecular reaction rates from a few characteristics of the potential surface.
Non-adiabatic dynamics consists of taking the interaction between several coupled potential energy surfaces (corresponding to different electronic quantum states of the molecule). The coupling terms are called vibronic couplings. The pioneering work in this field was done by Stueckelberg , Landau , and Zener in the 1930s, in their work on what is now known as the Landau–Zener transition . Their formula allows the transition probability between two adiabatic potential curves in the neighborhood of an avoided crossing to be calculated. Spin-forbidden reactions are one type of non-adiabatic reactions where at least one change in spin state occurs when progressing from reactant to product . | https://en.wikipedia.org/wiki/Quantum_chemistry |
Quantum chromodynamics binding energy ( QCD binding energy ), gluon binding energy or chromodynamic binding energy is the energy binding quarks together into hadrons . It is the energy of the field of the strong force , which is mediated by gluons . Motion-energy and interaction-energy contribute most of the hadron's mass. [ 1 ]
Most of the mass of hadrons is actually QCD binding energy, through mass–energy equivalence . This phenomenon is related to chiral symmetry breaking . In the case of nucleons — protons and neutrons — QCD binding energy forms about 99% of the nucleon's mass.
The kinetic energy of the hadron's constituents, moving at near the speed of light , contributes greatly to the hadron mass; [ 1 ] otherwise most of the rest is actual QCD binding energy, which emerges in a complex way from the potential-like terms in the QCD Lagrangian .
For protons, the sum of the rest masses of the three valence quarks (two up quarks and one down quark ) is approximately 9.4 MeV/ c 2 , while the proton's total mass is about 938.3 MeV/ c 2 . In the standard model, this "quark current mass" can nominally be attributed to the Higgs interaction . For neutrons, the sum of the rest masses of the three valence quarks (two down quarks and one up quark) is approximately 11.9 MeV/ c 2 , while the neutron's total mass is about 939.6 MeV/ c 2 . Considering that nearly all of the atom 's mass is concentrated in the nucleons, this means that about 99% of the mass of everyday matter ( baryonic matter ) is, in fact, chromodynamic binding energy.
While gluons are massless , they still possess energy — chromodynamic binding energy. In this way, they are similar to photons , which are also massless particles carrying energy — photon energy . The amount of energy per single gluon, or "gluon energy", cannot be directly measured, though a distribution can by inferred from deep inelastic scattering (DIS) experiments (see ref [4] for an old but still valid introduction.) Unlike photon energy, which is quantifiable, described by the Planck–Einstein relation and depends on a single variable (the photon's frequency ), no simple formula exists for the quantity of energy carried by each gluon. While the effects of a single photon can be observed, single gluons have not been observed outside of a hadron. A hadron is in totality [ 2 ] composed of gluons, valence quarks, sea quarks and other virtual particles .
The gluon content of a hadron can be inferred from DIS measurements. Again, not all of the QCD binding energy is gluon interaction energy, but rather, some of it comes from the kinetic energy of the hadron's constituents. [ 3 ] Currently, the total QCD binding energy per hadron can be estimated through a combination of the factors mentioned. In the future, studies into quark–gluon plasma will better complement the DIS studies and improve our understanding of the situation.
° Halzen, Francis and Martin, John, "Quarks and Leptons:An Introductory Course in Modem Particle Physics", John Wiley & Sons (1984). | https://en.wikipedia.org/wiki/Quantum_chromodynamics_binding_energy |
Consider two remote players, connected by a channel, that don't trust each other. The problem of them agreeing on a random bit by exchanging messages over this channel, without relying on any trusted third party, is called the coin flipping problem in cryptography. [ 1 ] Quantum coin flipping uses the principles of quantum mechanics to encrypt messages for secure communication. It is a cryptographic primitive which can be used to construct more complex and useful cryptographic protocols, [ 2 ] e.g. Quantum Byzantine agreement .
Unlike other types of quantum cryptography (in particular, quantum key distribution ), quantum coin flipping is a protocol used between two users who do not trust each other. [ 3 ] Consequently, both users (or players) want to win the coin toss and will attempt to cheat in various ways. [ 3 ]
In the classical setting, i.e. without quantum communication, one player can (in principle) always cheat against any protocol. [ 4 ] There are classical protocols based on commitment schemes , but they assume that the players lack the computing power to break the scheme. In contrast, quantum coin flipping protocols can resist cheating even by players with unlimited computing power.
The most basic figure of merit for a coin-flipping protocol is given by its bias, a number between 0 {\displaystyle 0} and 1 / 2 {\displaystyle 1/2} . The bias of a protocol captures the success probability of an all-powerful cheating player who uses the best conceivable strategy. A protocol with bias 0 {\displaystyle 0} means that no player can cheat. A protocol with bias 1 / 2 {\displaystyle 1/2} means that at least one player can always succeed at cheating. Obviously, the smaller the bias better the protocol.
When the communication is over a quantum channel , it has been shown that even the best conceivable protocol can not have a bias less than 1 / 2 − 1 / 2 ≈ 0.2071 {\displaystyle 1/{\sqrt {2}}-1/2\approx 0.2071} . [ 5 ] [ 6 ]
Consider the case where each player knows the preferred bit of the other. A coin flipping problem which makes this additional assumption constitutes the weaker variant thereof called weak coin flipping (WCF). In the case of classical channels this extra assumption yields no improvement. On the other hand, it has been proven that WCF protocols with arbitrarily small biases do exist. [ 7 ] [ 8 ] However, the best known explicit WCF protocol has bias 1 / 6 ≈ 0.1667 {\displaystyle 1/6\approx 0.1667} . [ 9 ]
Although quantum coin flipping offers clear advantages over its classical counterpart in theory, accomplishing it in practice has proven difficult. [ 3 ] [ 10 ]
Manuel Blum introduced coin flipping as part of a classical system in 1983 based on computational algorithms and assumptions. [ 11 ] Blum's version of coin flipping answers the following cryptographic problem:
Thus, the problem with Alice and Bob is that they do not trust each other; the only resource they have is the telephone communication channel, and there is not a third party available to read the coin. Therefore, Alice and Bob must be either truthful and agree on a value or be convinced that the other is cheating. [ 12 ]
In 1984, quantum cryptography emerged from a paper written by Charles H. Bennett and Giles Brassard. In this paper, the two introduced the idea of using quantum mechanics to enhance previous cryptographic protocols such as coin flipping. [ 3 ] Since then, many researchers have applied quantum mechanics to cryptography as they have proven theoretically to be more secure than classical cryptography, however, demonstrating these protocols in practical systems is difficult to accomplish.
As published in 2014, a group of scientists at the Laboratory for Communication and Processing of Information (LTCI) in Paris have implemented quantum coin flipping protocols experimentally. [ 3 ] The researchers have reported that the protocol performs better than a classical system over a suitable distance for a metropolitan area optical network. [ 3 ]
In cryptography, coin flipping is defined to be the problem where two mutually distrustful and remote players want to agree on a random bit without relying on any third party. [ 1 ]
In quantum cryptography, strong coin flipping (SCF) is defined to be a coin flipping problem where each player is oblivious to the preference of the other. [ 13 ]
In quantum cryptography, weak coin flipping (WCF) is defined to be a coin flipping problem where each player knows the preference of the other. [ 14 ]
It follows that the players have opposite preferences. If this were not the case then the problem will be pointless as the players can simply choose the outcome they desire.
Consider any coin flipping protocol. Let Alice and Bob be the two players who wish to implement the protocol. Consider the scenario where Alice cheats using her best strategy against Bob who honestly follows the protocol. Let the probability that Bob obtains the outcome Alice preferred be given by P A ∗ {\displaystyle P_{A}^{*}} . Consider the reversed situation, i.e. Bob cheats using his best strategy against Alice who honestly follows the protocol. Let the corresponding probability that Alice obtains the outcome Bob preferred to be given by P B ∗ {\displaystyle P_{B}^{*}} .
The bias of the protocol is defined to be ϵ := max [ P A ∗ , P B ∗ ] − 1 2 {\textstyle \epsilon :=\max[P_{A}^{*},P_{B}^{*}]-{\frac {1}{2}}} .
The half is subtracted because a player will get the desired value half the time purely by chance.
Coin flipping can be defined for biased coins as well, i.e. the bits are not equally likely. The notion of correctness has also been formalized which requires that when both players follow the protocol (nobody cheats) the players always agree on the bit generated and that the bit follows some fixed probability distribution.
Quantum coin flipping and other types of quantum cryptography communicate information through the transmission of qubits . The accepting player does not know the information in the qubit until he performs a measurement. [ 12 ] Information about each qubit is stored on and carried by a single photon . [ 10 ] Once the receiving player measures the photon, it is altered, and will not produce the same output if measured again. [ 10 ] Since a photon can only be read the same way once, any other party attempting to intercept the message is easily detectable. [ 10 ]
Quantum coin flipping is when random qubits are generated between two players that do not trust each other because both of them want to win the coin toss, which could lead them to cheat in a variety of ways. [ 3 ] The essence of coin flipping occurs when the two players issue a sequence of instructions over a communication channel that then eventually results in an output. [ 10 ]
A basic quantum coin flipping protocol involves two people: Alice and Bob. [ 11 ]
A more general explanation of the above protocol is as follows: [ 15 ]
There are a few assumptions that must be made for this protocol to work properly. The first is that Alice can create each state independent of Bob, and with an equal probability. Second, for the first bit that Bob successfully measures, his basis and bit are both random and completely independent of Alice. The last assumption, is that when Bob measures a state, he has a uniform probability to measure each state, and no state is easier to be detected than others. This last assumption is especially important because if Alice were aware of Bob's inability to measure certain states, she could use that to her advantage. [ 11 ]
The key issue with coin flipping is that it occurs between two distrustful parties. [ 15 ] These two parties are communicating through the communication channel some distance from each other and they have to agree on a winner or loser with each having a 50 percent chance of winning. [ 15 ] However, since they are distrustful of one another, cheating is likely to occur. Cheating can occur in a number of ways such as claiming they lost some of the message when they do not like the result or increasing the average number of photons contained in each of the pulses. [ 3 ]
For Bob to cheat, he would have to be able to guess Alice's basis with a probability greater than 1 / 2 . [ 15 ] In order to accomplish this, Bob would have to be able to determine a train of photons randomly polarized in one basis from a train of photons polarized in another basis. [ 15 ]
Alice, on the other hand, could cheat in a couple of different ways, but she has to be careful because Bob could easily detect it. [ 15 ] When Bob sends a correct guess to Alice, she could convince Bob that her photons are actually polarized the opposite of Bob's correct guess. [ 15 ] Alice could also send Bob a different original sequence than she actually used in order to beat Bob. [ 15 ]
Single photons are used to pass the information from one player to the other (qubits). [ 10 ] In this protocol, the information is encoded in the single photons with polarization directions of 0, 45, 90, and 135 degrees, non-orthogonal quantum states. [ 15 ] When a third party attempts to read or gain information on the transmission, they alter the photon's polarization in a random way that is likely detected by the two players because it does not match the pattern exchanged between the two legitimate users. [ 15 ]
The Dip Dip Boom (DDB) protocol is a quantum version of the following game. [ 9 ] Consider a list of numbers p i {\displaystyle {p_{i}}} , each between 0 and 1. The players, Alice and Bob, take turns to say "Dip" or "Boom" with probability p i {\displaystyle p_{i}} at round i {\displaystyle i} . The player who says "Boom" wins. Obviously, a cheating player can simply say "Boom" and win as there are no rewards for longer games. We will consider games that terminate so that for some (large) i {\displaystyle i} , say n {\displaystyle n} , we set p i = 1 {\displaystyle p_{i}=1} .
Consider round i {\displaystyle i} . Let us denote by P A ( i ) {\displaystyle P_{A}(i)} and P B ( i ) {\displaystyle P_{B}(i)} the probability of, respectively, Alice and Bob winning. Let P U ( i ) {\displaystyle P_{U}(i)} be the probability that the game remains undecided. These numbers for the classical game described above can be evaluated inductively.
We now describe the quantum version. Let A , B {\displaystyle {\mathcal {A}},{\mathcal {B}}} be a three dimensional Hilbert space spanned by | A ⟩ , | B ⟩ , | U ⟩ {\displaystyle |A\rangle ,|B\rangle ,|U\rangle } . Let M {\displaystyle {\mathcal {M}}} be a two dimensional Hilbert space which is spanned by | DIP ⟩ , | BOOM ⟩ {\displaystyle |{\text{DIP}}\rangle ,|{\text{BOOM}}\rangle } .
It has been shown that using a WCF protocol with an arbitrarily small bias one can construct a SCF protocol with bias arbitrarily close to 1 2 − 1 2 {\textstyle {\frac {1}{\sqrt {2}}}-{\frac {1}{2}}} which is known to be optimal. [ 16 ]
As mentioned in the history section, scientists at the LTCI in Paris have experimentally carried out a quantum coin flipping protocol. Previous protocols called for a single photon source or an entangled source to be secure. However, these sources are why it is difficult for quantum coin flipping to be implemented. Instead, the researchers at LTCI used the effects of quantum superposition rather than a single photon source, which they claim makes implementation easier with the standard photon sources available. [ 3 ]
The researchers used the Clavis2 platform developed by IdQuantique for their protocol, but needed to modify the Clavis2 system in order for it to work for the coin flipping protocol. The experimental setup they used with the Clavis2 system, involves a two-way approach. Light pulsed at 1550 nanometres is sent from Bob to Alice. Alice then uses a phase modulator to encrypt the information. After encryption, she then uses a Faraday mirror to reflect and attenuate the pulses at her chosen level and sends them back to Bob. Using two high quality single photon detectors, Bob chooses a measurement basis in his phase modulator to detect the pulses from Alice. [ 11 ]
They replaced the detectors on Bob's side because of the low detection efficiencies of the previous detectors. When they replaced the detectors, they were able to show a quantum advantage on a channel for over 15 kilometres (9.3 mi). A couple of other challenges the group faced was reprogramming the system because photon source attenuation was high and performing system analyses to identify losses and errors in system components. With these corrections, the scientists were capable of implementing a coin flipping protocol by introducing a small honest abort probability, the probability that two honest participants cannot obtain a coin flip at the end of the protocol, but at a short communication distance. [ 3 ] | https://en.wikipedia.org/wiki/Quantum_coin_flipping |
The terminology quantum compass often relates to an instrument which measures relative position using the technique of atom interferometry . It includes an ensemble of accelerometers and gyroscope based on quantum technology [ 1 ] to form an inertial navigation unit.
Work on quantum technology based inertial measurement units ( IMUs ), the instruments containing the gyroscopes and accelerometers, follows from early demonstrations of matter-wave based accelerometers and gyrometers. [ 2 ] The first demonstration of onboard acceleration measurement was made on an Airbus A300 in 2011. [ 3 ]
A quantum compass contains clouds of atoms frozen using lasers . By measuring the movement of these frozen particles over precise periods of time the motion of the device can be calculated. The device would then provide accurate position in circumstances where satellites are not available for satellite navigation , e.g. a fully submerged submarine. [ 4 ]
Various defence agencies worldwide, such as the US DARPA [ 5 ] and the United Kingdom Ministry of Defence [ 6 ] [ 4 ] have pushed the development of prototypes for future uses in submarines and aircraft.
In 2024, researchers from the Centre for Cold Matter of Imperial College , London, tested an experimental quantum compass on an underground train on London's District line . [ 7 ] During the same year, scientists at the Sandia National Laboratories announced they were able to perform spatial quantum sensing using silicon photonic microchip components, a significant advancement towards the development of compact, portable and inexpensive quantum compass devices. [ 8 ]
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it .
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_compass |
Quantum computational chemistry is an emerging field that exploits quantum computing to simulate chemical systems. Despite quantum mechanics' foundational role in understanding chemical behaviors, traditional computational approaches face significant challenges, largely due to the complexity and computational intensity of quantum mechanical equations. This complexity arises from the exponential growth of a quantum system's wave function with each added particle, making exact simulations on classical computers inefficient. [ 1 ]
Efficient quantum algorithms for chemistry problems are expected to have run-times and resource requirements that scale polynomially with system size and desired accuracy. Experimental efforts have validated proof-of-principle chemistry calculations, though currently limited to small systems. [ 1 ]
While there are several common methods in quantum chemistry , the section below lists only a few examples.
Qubitization is a mathematical and algorithmic concept in quantum computing for the simulation of quantum systems via Hamiltonian dynamics. The core idea of qubitization is to encode the problem of Hamiltonian simulation in a way that is more efficiently processable by quantum algorithms. [ 4 ]
Qubitization involves a transformation of the Hamiltonian operator, a central object in quantum mechanics representing the total energy of a system. In classical computational terms, a Hamiltonian can be thought of as a matrix describing the energy interactions within a quantum system. The goal of qubitization is to embed this Hamiltonian into a larger, unitary operator, which is a type of operator in quantum mechanics that preserves the norm of vectors upon which it acts. [ 4 ]
Mathematically, the process of qubitization constructs a unitary operator U {\displaystyle U} such that a specific projection of U {\displaystyle U} is proportional to the Hamiltonian H {\displaystyle H} of interest. This relationship can often be represented as H = ⟨ G | U | G ⟩ {\displaystyle H=\langle G|U|G\rangle } , where | G ⟩ {\displaystyle |G\rangle } is a specific quantum state and ⟨ G | {\displaystyle \langle G|} is its conjugate transpose . The efficiency of this method comes from the fact that the unitary operator U {\displaystyle U} can be implemented on a quantum computer with fewer resources (like qubits and quantum gates) than would be required for directly simulating H . {\displaystyle H.} [ 4 ]
A key feature of qubitization is in simulating Hamiltonian dynamics with high precision while reducing the quantum resource overhead. This efficiency is especially beneficial in quantum algorithms where the simulation of complex quantum systems is necessary, such as in quantum chemistry and materials science simulations. Qubitization also develops quantum algorithms for solving certain types of problems more efficiently than classical algorithms. For instance, it has implications for the Quantum Phase Estimation algorithm, which is fundamental in various quantum computing applications, including factoring and solving linear systems of equations. [ 4 ]
In Gaussian orbital basis sets , phase estimation algorithms have been optimized empirically from O ( M 11 ) {\displaystyle {\mathcal {O}}(M^{11})} to O ( M 5 ) {\displaystyle {\mathcal {O}}(M^{5})} where M {\displaystyle M} is the number of basis sets. Advanced Hamiltonian simulation algorithms have further reduced the scaling, with the introduction of techniques like Taylor series methods and qubitization, providing more efficient algorithms with reduced computational requirements. [ 5 ]
Plane wave basis sets, suitable for periodic systems, have also seen advancements in algorithm efficiency, with improvements in product formula-based approaches and Taylor series methods. [ 4 ]
Phase estimation, as proposed by Kitaev in 1996, identifies the lowest energy eigenstate ( | E 0 ⟩ {\displaystyle |E_{0}\rangle } ) and excited states ( | E i ⟩ {\displaystyle |E_{i}\rangle } ) of a physical Hamiltonian, as detailed by Abrams and Lloyd in 1999. [ 6 ] In quantum computational chemistry, this technique is employed to encode fermionic Hamiltonians into a qubit framework. [ 7 ]
The qubit register is initialized in a state, which has a nonzero overlap with the Full Configuration Interaction (FCI) target eigenstate of the system. This state | ψ ⟩ {\displaystyle |\psi \rangle } is expressed as a sum over the energy eigenstates of the Hamiltonian, | ψ ⟩ = ∑ i = 1 c i | E i ⟩ {\displaystyle |\psi \rangle =\sum _{i=1}c_{i}|E_{i}\rangle } , where c i {\displaystyle c_{i}} represents complex coefficients. [ 9 ]
Each ancilla qubit undergoes a Hadamard gate application, placing the ancilla register in a superposed state. Subsequently, controlled gates, as shown above, modify this state. [ 9 ]
This transform is applied to the ancilla qubits, revealing the phase information that encodes the energy eigenvalues. [ 9 ]
The ancilla qubits are measured in the Z basis, collapsing the main register into the corresponding energy eigenstate | E i ⟩ {\displaystyle |E_{i}\rangle } based on the probability | c i | 2 {\displaystyle |c_{i}|^{2}} . [ 9 ]
The algorithm requires ω {\displaystyle \omega } ancilla qubits, with their number determined by the desired precision and success probability of the energy estimate. Obtaining a binary energy estimate precise to n bits with a success probability p {\displaystyle p} necessitates. [ 9 ] ω = n + ⌈ log 2 ( 2 + 1 2 p ) ⌉ {\displaystyle \omega =n+\lceil \log _{2}\left(2+{\frac {1}{2p}}\right)\rceil } ancilla qubits. This phase estimation has been validated experimentally across various quantum architectures. [ 9 ]
The total coherent time evolution T {\displaystyle T} required for the algorithm is approximately T = 2 ( ω + 1 ) π {\displaystyle T=2^{(\omega +1)}\pi } . [ 10 ] The total evolution time is related to the binary precision ε PE = 1 2 n {\displaystyle \varepsilon _{\text{PE}}={\frac {1}{2^{n}}}} , with an expected repeat of the procedure for accurate ground state estimation. Errors in the algorithm include errors in energy eigenvalue estimation ( ε P E {\displaystyle \varepsilon _{PE}} ), unitary evolutions ( ε U {\displaystyle \varepsilon _{U}} ), and circuit synthesis errors ( ε C S {\displaystyle \varepsilon _{CS}} ), which can be quantified using techniques like the Solovay-Kitaev theorem . [ 11 ]
The phase estimation algorithm can be enhanced or altered in several ways, such as using a single ancilla qubit for sequential measurements, increasing efficiency, parallelization, or enhancing noise resilience in analytical chemistry. The algorithm can also be scaled using classically obtained knowledge about energy gaps between states. [ 12 ]
Effective state preparation is needed, as a randomly chosen state would exponentially decrease the probability of collapsing to the desired ground state. Various methods for state preparation have been proposed, including classical approaches and quantum techniques like adiabatic state preparation. [ 13 ]
The variational quantum eigensolver is an algorithm in quantum computing, crucial for near-term quantum hardware. [ 14 ] Initially proposed by Peruzzo et al. in 2014 and further developed by McClean et al. in 2016, VQE finds the lowest eigenvalue of Hamiltonians, particularly those in chemical systems. [ 15 ] It employs the variational method (quantum mechanics) , which guarantees that the expectation value of the Hamiltonian for any parameterized trial wave function is at least the lowest energy eigenvalue of that Hamiltonian. [ 16 ] VQE is a hybrid algorithm that utilizes both quantum and classical computers. The quantum computer prepares and measures the quantum state, while the classical computer processes these measurements and updates the system. This synergy allows VQE to overcome some limitations of purely quantum methods. [ 17 ]
The reduced density matrices (1-RDM and 2-RDM) can be used to extrapolate the electronic structure of a system. [ 18 ]
In the Hamiltonian variational ansatz, the initial state | ψ 0 ⟩ {\displaystyle |\psi _{0}\rangle } is prepared to represent the ground state of the molecular Hamiltonian without electron correlations. The evolution of this state under the Hamiltonian, split into commuting segments H j {\displaystyle H_{j}} , is given by the equation below. [ 17 ]
| ψ ( θ ) ⟩ = ∏ d ∏ j e i θ d , j H j | ψ 0 ⟩ {\displaystyle |\psi (\theta )\rangle =\prod _{d}\prod _{j}e^{i\theta _{d,j}H_{j}}|\psi _{0}\rangle }
where θ d , j {\displaystyle \theta _{d,j}} are variational parameters optimized to minimize the energy, providing insights into the electronic structure of the molecule. [ 17 ]
McClean et al. (2016) and Romero et al. (2019) proposed a formula to estimate the number of measurements ( N m {\displaystyle N_{m}} ) required for energy precision. The formula is given by N m = ( ∑ i | h i | ) 2 / ϵ 2 {\displaystyle N_{m}=\left(\sum _{i}|h_{i}|\right)^{2}/\epsilon ^{2}} , where h i {\displaystyle h_{i}} are coefficients of each Pauli string in the Hamiltonian. This leads to a scaling of O ( M 6 / ϵ 2 ) {\displaystyle {\mathcal {O}}(M^{6}/\epsilon ^{2})} in a Gaussian orbital basis and O ( M 4 / ϵ 2 ) {\displaystyle {\mathcal {O}}(M^{4}/\epsilon ^{2})} in a plane wave dual basis. Note that M {\displaystyle M} is the number of basis functions in the chosen basis set. [ 19 ] [ 20 ]
A method by Bonet-Monroig, Babbush, and O'Brien (2019) focuses on grouping terms at a fermionic level rather than a qubit level, leading to a measurement requirement of only O ( M 2 ) {\displaystyle {\mathcal {O}}(M^{2})} circuits with an additional gate depth of O ( M ) {\displaystyle {\mathcal {O}}(M)} . [ 21 ]
While VQE's application in solving the electronic Schrödinger equation for small molecules has shown success, its scalability is hindered by two main challenges: the complexity of the quantum circuits required and the intricacies involved in the classical optimization process. [ 22 ] These challenges are significantly influenced by the choice of the variational ansatz, which is used to construct the trial wave function. Modern quantum computers face limitations in running deep quantum circuits, especially when using the existing ansatzes for problems that exceed several qubits. [ 17 ]
Jordan-Wigner encoding is a method in quantum computing used for simulating fermionic systems like molecular orbitals and electron interactions in quantum chemistry. [ 23 ]
In quantum chemistry, electrons are modeled as fermions with antisymmetric wave functions . The Jordan-Wigner encoding maps these fermionic orbitals to qubits, preserving their antisymmetric nature. Mathematically, this is achieved by associating each fermionic creation ( a i † ) {\displaystyle (a_{i}^{\dagger })} and annihilation ( a i ) {\displaystyle (a_{i})} operator with corresponding qubit operators through the Jordan-Wigner transformation :
a i † → 1 2 ( ∏ k = 1 i − 1 Z k ) ( X i − i Y i ) {\displaystyle a_{i}^{\dagger }\rightarrow {\frac {1}{2}}\left(\prod _{k=1}^{i-1}Z_{k}\right)(X_{i}-iY_{i})}
Where X i {\displaystyle X_{i}} , Y i {\displaystyle Y_{i}} , and Z i {\displaystyle Z_{i}} are Pauli matrices acting on the i th {\displaystyle i^{\text{th}}} qubit. [ 23 ]
Electron hopping between orbitals, central to chemical bonding and reactions, is represented by terms like a † i a j + a † j a i {\displaystyle a^{\dagger _{i}}a_{j}+a^{\dagger _{j}}a_{i}} . Under Jordan-Wigner encoding, these transform as follows: [ 23 ] a i † a j + a j † a i → 1 2 ( X i X j + Y i Y j ) Z i + 1 ⋯ Z j − 1 {\displaystyle a_{i}^{\dagger }a_{j}+a_{j}^{\dagger }a_{i}\rightarrow {\frac {1}{2}}(X_{i}X_{j}+Y_{i}Y_{j})Z_{i+1}\cdots Z_{j-1}} This transformation captures the quantum mechanical behavior of electron movement and interaction within molecules. [ 24 ]
The complexity of simulating a molecular system using Jordan-Wigner encoding is influenced by the structure of the molecule and the nature of electron interactions. For a molecular system with K {\displaystyle K} orbitals, the number of required qubits scales linearly with K {\displaystyle K} , but the complexity of gate operations depends on the specific interactions being modeled. [ 25 ]
The Jordan-Wigner transformation encodes fermionic operators into qubit operators, but it introduces non-local string operators that can make simulations inefficient. The FSWAP gate is used to mitigate this inefficiency by rearranging the ordering of fermions (or their qubit representations), thus simplifying the implementation of fermionic operations. [ 26 ]
FSWAP networks rearrange qubits to efficiently simulate electron dynamics in molecules. These networks are essential for reducing the gate complexity in simulations, especially for non-neighboring electron interactions . [ 27 ]
When two fermionic modes (represented as qubits after the Jordan-Wigner transformation) are swapped, the FSWAP gate not only exchanges their states but also correctly updates the phase of the wavefunction to maintain fermionic antisymmetry . This is in contrast to the standard SWAP gate , which does not account for the phase change required in the antisymmetric wavefunctions of fermions. [ 28 ]
The use of FSWAP gates can significantly reduce the complexity of quantum circuits for simulating fermionic systems. By intelligently rearranging the fermions, the number of gates required to simulate certain fermionic operations can be reduced, leading to more efficient simulations. This is particularly useful in simulations where fermions need to be moved across large distances within the system, as it can avoid the need for long chains of operations that would otherwise be required. [ 29 ] | https://en.wikipedia.org/wiki/Quantum_computational_chemistry |
The quantum concentration n Q is the particle concentration (i.e. the number of particles per unit volume) of a system where the interparticle distance is equal to the thermal de Broglie wavelength .
Quantum effects become appreciable when the particle concentration is greater than or equal to the quantum concentration, which is defined as: [ 1 ]
where:
The quantum concentration for room temperature protons is about 1/cubic-Angstrom.
As the quantum concentration depends on temperature, high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf .
For an ideal gas the Sackur–Tetrode equation can be written in terms of the quantum concentration as [ 1 ]
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This article about statistical mechanics is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_concentration |
Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured (the measurement context). More formally, the measurement result (assumed pre-existing) of a quantum observable is dependent upon which other commuting observables are within the same measurement set.
Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem . [ 1 ] [ 2 ] The study of contextuality has developed into a major topic of interest in quantum foundations as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory, [ 3 ] graph theory , [ 4 ] hypergraphs , [ 5 ] algebraic topology , [ 6 ] and probabilistic couplings . [ 7 ]
Nonlocality , in the sense of Bell's theorem , may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over spacelike separated regions. This follows from Fine's theorem. [ 8 ] [ 3 ]
Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing . [ 9 ] [ 10 ] [ 11 ] [ 12 ] Contemporary research has increasingly focused on exploring its utility as a computational resource.
The need for contextuality was discussed informally in 1935 by Grete Hermann , [ 13 ] but it was more than 30 years later when Simon B. Kochen and Ernst Specker , and separately John Bell , constructed proofs that any realistic hidden-variable theory able to explain the phenomenology of quantum mechanics is contextual for systems of Hilbert space dimension three and greater. The Kochen–Specker theorem proves that realistic noncontextual hidden-variable theories cannot reproduce the empirical predictions of quantum mechanics. [ 14 ] Such a theory would suppose the following.
In addition, Kochen and Specker constructed an explicitly noncontextual hidden-variable model for the two-dimensional qubit case in their paper on the subject, [ 1 ] thereby completing the characterisation of the dimensionality of quantum systems that can demonstrate contextual behaviour. Bell's proof invoked a weaker version of Gleason's theorem , reinterpreting the theorem to show that quantum contextuality exists only in Hilbert space dimension greater than two. [ 2 ]
The sheaf -theoretic, or Abramsky–Brandenburger, approach to contextuality initiated by Samson Abramsky and Adam Brandenburger is theory-independent and can be applied beyond quantum theory to any situation in which empirical data arises in contexts. As well as being used to study forms of contextuality arising in quantum theory and other physical theories, it has also been used to study formally equivalent phenomena in logic , [ 15 ] relational databases , [ 16 ] natural language processing , [ 17 ] and constraint satisfaction . [ 18 ]
In essence, contextuality arises when empirical data is locally consistent but globally inconsistent .
This framework gives rise in a natural way to a qualitative hierarchy of contextuality:
Each level in this hierarchy strictly includes the next. An important intermediate level that lies strictly between the logical and strong contextuality classes is all-versus-nothing contextuality , [ 15 ] a representative example of which is the Greenberger–Horne–Zeilinger proof of nonlocality.
Adán Cabello, Simone Severini , and Andreas Winter introduced a general graph-theoretic framework for studying contextuality of different physical theories. [ 19 ] Within this framework experimental scenarios are described by graphs, and certain invariants of these graphs were shown have particular physical significance. One way in which contextuality may be witnessed in measurement statistics is through the violation of noncontextuality inequalities (also known as generalized Bell inequalities). With respect to certain appropriately normalised inequalities, the independence number , Lovász number , and fractional packing number of the graph of an experimental scenario provide tight upper bounds on the degree to which classical theories, quantum theory, and generalised probabilistic theories, respectively, may exhibit contextuality in an experiment of that kind. A more refined framework based on hypergraphs rather than graphs is also used. [ 5 ]
In the CbD approach, [ 20 ] [ 21 ] [ 22 ] developed by Ehtibar Dzhafarov, Janne Kujala, and colleagues, (non)contextuality is treated as a property of any system of random variables , defined as a set R = { R q c : q ∈ Q , q ≺ c , c ∈ C } {\displaystyle {\mathcal {R}}=\{R_{q}^{c}:q\in Q,q\prec c,c\in C\}} in which each random variable R q c {\displaystyle R_{q}^{c}} is labeled by its content q {\displaystyle q} – the property it measures, and its context c {\displaystyle c} – the set of recorded circumstances under which it is recorded (including but not limited to which other random variables it is recorded together with); q ≺ c {\displaystyle q\prec c} stands for " q {\displaystyle q} is measured in c {\displaystyle c} ". The variables within a context are jointly distributed, but variables from different contexts are stochastically unrelated , defined on different sample spaces. A (probabilistic) coupling of the system R {\displaystyle {\mathcal {R}}} is defined as a system S {\displaystyle S} in which all variables are jointly distributed and, in any context c {\displaystyle c} , R c = { R q c : q ∈ Q , q ≺ c } {\displaystyle R^{c}=\{R_{q}^{c}:q\in Q,q\prec c\}} and S c = { S q c : q ∈ Q , q ≺ c } {\displaystyle S^{c}=\{S_{q}^{c}:q\in Q,q\prec c\}} are identically distributed. The system is considered noncontextual if it has a coupling S {\displaystyle S} such that the probabilities Pr [ S q c = S q c ′ ] {\displaystyle \Pr[S_{q}^{c}=S_{q}^{c'}]} are maximal possible for all contexts c , c ′ {\displaystyle c,c'} and contents q {\displaystyle q} such that q ≺ c , c ′ {\displaystyle q\prec c,c'} . If such a coupling does not exist, the system is contextual. For the important class of cyclic systems of dichotomous ( ± 1 {\displaystyle \pm 1} ) random variables, C n = { ( R 1 1 , R 2 1 ) , ( R 2 2 , R 3 2 ) , … , ( R n n , R 1 n ) } {\displaystyle {\mathcal {C}}_{n}={\big \{}(R_{1}^{1},R_{2}^{1}),(R_{2}^{2},R_{3}^{2}),\ldots ,(R_{n}^{n},R_{1}^{n}){\big \}}} ( n ≥ 2 {\displaystyle n\geq 2} ), it has been shown [ 23 ] [ 24 ] that such a system is noncontextual if and only if D ( C n ) ≤ Δ ( C n ) , {\displaystyle D({\mathcal {C}}_{n})\leq \Delta ({\mathcal {C}}_{n}),} where Δ ( C n ) = ( n − 2 ) + | R 1 1 − R 1 n | + | R 2 1 − R 2 2 | + … + | R n n − 1 − R n n | , {\displaystyle \Delta ({\mathcal {C}}_{n})=(n-2)+|R_{1}^{1}-R_{1}^{n}|+|R_{2}^{1}-R_{2}^{2}|+\ldots +|R_{n}^{n-1}-R_{n}^{n}|,} and D ( C n ) = max ( λ 1 ⟨ R 1 1 R 2 1 ⟩ + λ 2 ⟨ R 2 2 R 3 2 ⟩ + … + λ n ⟨ R n n R 1 n ⟩ ) , {\displaystyle D({\mathcal {C}}_{n})=\max {\big (}\lambda _{1}\langle R_{1}^{1}R_{2}^{1}\rangle +\lambda _{2}\langle R_{2}^{2}R_{3}^{2}\rangle +\ldots +\lambda _{n}\langle R_{n}^{n}R_{1}^{n}\rangle {\big )},} with the maximum taken over all λ i = ± 1 {\displaystyle \lambda _{i}=\pm 1} whose product is − 1 {\displaystyle -1} . If R q c {\displaystyle R_{q}^{c}} and R q c ′ {\displaystyle R_{q}^{c'}} , measuring the same content in different context, are always identically distributed, the system is called consistently connected (satisfying "no-disturbance" or "no-signaling" principle). Except for certain logical issues, [ 7 ] [ 21 ] in this case CbD specializes to traditional treatments of contextuality in quantum physics. In particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to D ( C n ) ≤ n − 2 , {\displaystyle D({\mathcal {C}}_{n})\leq n-2,} which includes the Bell/CHSH inequality ( n = 4 {\displaystyle n=4} ), KCBS inequality ( n = 5 {\displaystyle n=5} ), and other famous inequalities. [ 25 ] That nonlocality is a special case of contextuality follows in CbD from the fact that being jointly distributed for random variables is equivalent to being measurable functions of one and the same random variable (this generalizes Arthur Fine 's analysis of Bell's theorem ). CbD essentially coincides with the probabilistic part of Abramsky's sheaf-theoretic approach if the system is strongly consistently connected , which means that the joint distributions of { R q 1 c , … , R q k c } {\displaystyle \{R_{q_{1}}^{c},\ldots ,R_{q_{k}}^{c}\}} and { R q 1 c ′ , … , R q k c ′ } {\displaystyle \{R_{q_{1}}^{c'},\ldots ,R_{q_{k}}^{c'}\}} coincide whenever q 1 , … , q k {\displaystyle q_{1},\ldots ,q_{k}} are measured in contexts c , c ′ {\displaystyle c,c'} . However, unlike most approaches to contextuality, CbD allows for inconsistent connectedness , with R q c {\displaystyle R_{q}^{c}} and R q c ′ {\displaystyle R_{q}^{c'}} differently distributed. This makes CbD applicable to physics experiments in which no-disturbance condition is violated, [ 24 ] [ 26 ] as well as to human behavior where this condition is violated as a rule. [ 27 ] In particular, Vctor Cervantes, Ehtibar Dzhafarov, and colleagues have demonstrated that random variables describing certain paradigms of simple decision making form contextual systems, [ 28 ] [ 29 ] [ 30 ] whereas many other decision-making systems are noncontextual once their inconsistent connectedness is properly taken into account. [ 27 ]
An extended notion of contextuality due to Robert Spekkens applies to preparations and transformations as well as to measurements, within a general framework of operational physical theories. [ 31 ] With respect to measurements, it removes the assumption of determinism of value assignments that is present in standard definitions of contextuality. This breaks the interpretation of nonlocality as a special case of contextuality, and does not treat irreducible randomness as nonclassical. Nevertheless, it recovers the usual notion of contextuality when outcome determinism is imposed.
Spekkens' contextuality can be motivated using Leibniz's law of the identity of indiscernibles . The law applied to physical systems in this framework mirrors the entended definition of noncontextuality. This was further explored by Simmons et al , [ 32 ] who demonstrated that other notions of contextuality could also be motivated by Leibnizian principles, and could be thought of as tools enabling ontological conclusions from operational statistics.
Given a pure quantum state | ψ ⟩ {\displaystyle |\psi \rangle } , Born's rule tells that the probability to obtain another state | ϕ ⟩ {\displaystyle |\phi \rangle } in a measurement is | ⟨ ϕ | ψ ⟩ | 2 {\displaystyle |\langle \phi |\psi \rangle |^{2}} . However, such a number does not define a full probability distribution, i.e. values over a set of mutually exclusive events, summing up to 1. In order to obtain such a set one needs to specify a context, that is a complete set of commuting operators (CSCO), or equivalently a set of N orthogonal projectors | ϕ n ⟩ ⟨ ϕ n | {\displaystyle |\phi _{n}\rangle \langle \phi _{n}|} that sum to identity, where N {\displaystyle N} is the dimension of the Hilbert space. Then one has ∑ n | ⟨ ϕ n | ψ ⟩ | 2 = 1 {\displaystyle \sum _{n}|\langle \phi _{n}|\psi \rangle |^{2}=1} as expected. In that sense, one can tell that a state vector | ψ ⟩ {\displaystyle |\psi \rangle } alone is predictively incomplete, as long a context has not been specified. [ 33 ] The actual physical state, now defined by | ϕ n ⟩ {\displaystyle |\phi _{n}\rangle } within a specified context, has been called a modality by Auffèves and Grangier [ 34 ] [ 35 ]
Since it is clear that | ψ ⟩ {\displaystyle |\psi \rangle } alone does not define a modality, what is its status ? If N ≥ 3 {\displaystyle N\geq 3} , one sees easily that | ψ ⟩ {\displaystyle |\psi \rangle } is associated with an equivalence class of modalities, belonging to different contexts, but connected between themselves with certainty, even if the different CSCO observables do not commute. This equivalence class is called an extravalence class, and the associated transfer of certainty between contexts is called extracontextuality. As a simple example, the usual singlet state for two spins 1/2 can be found in the (non commuting) CSCOs associated with the measurement of the total spin (with S = 0 , m = 0 {\displaystyle S=0,\;m=0} ), or with a Bell measurement, and actually it appears in infinitely many different CSCOs - but obviously not in all possible ones. [ 36 ]
The concepts of extravalence and extracontextuality are very useful to spell out the role of contextuality in quantum mechanics, that is not non-contextual (like classical physical would be), but not either fully contextual, since modalities belonging to incompatible (non-commuting) contexts may be connected with certainty. Starting now from extracontextuality as a postulate, the fact that certainty can be transferred between contexts, and is then associated with a given projector, is the very basis of the hypotheses of Gleason's theorem, and thus of Born's rule. [ 37 ] [ 38 ] Also, associating a state vector with an extravalence class clarifies its status as a mathematical tool to calculate probabilities connecting modalities, which correspond to the actual observed physical events or results. This point of view is quite useful, and it can be used everywhere in quantum mechanics.
A form of contextuality that may present in the dynamics of a quantum system was introduced by Shane Mansfield and Elham Kashefi , and has been shown to relate to computational quantum advantages . [ 39 ] As a notion of contextuality that applies to transformations it is inequivalent to that of Spekkens. Examples explored to date rely on additional memory constraints which have a more computational than foundational motivation. Contextuality may be traded-off against Landauer erasure to obtain equivalent advantages. [ 40 ]
The Kochen–Specker theorem proves that quantum mechanics is incompatible with realistic noncontextual hidden variable models. On the other hand Bell's theorem proves that quantum mechanics is incompatible with factorisable hidden variable models in an experiment in which measurements are performed at distinct spacelike separated locations. Arthur Fine showed that in the experimental scenario in which the famous CHSH inequalities and proof of nonlocality apply, a factorisable hidden variable model exists if and only if a noncontextual hidden variable model exists. [ 8 ] This equivalence was proven to hold more generally in any experimental scenario by Samson Abramsky and Adam Brandenburger . [ 3 ] It is for this reason that we may consider nonlocality to be a special case of contextuality.
A number of methods exist for quantifying contextuality. One approach is by measuring the degree to which some particular noncontextuality inequality is violated, e.g. the KCBS inequality , the Yu–Oh inequality, [ 41 ] or some Bell inequality . A more general measure of contextuality is the contextual fraction. [ 11 ]
Given a set of measurement statistics e , consisting of a probability distribution over joint outcomes for each measurement context, we may consider factoring e into a noncontextual part e NC and some remainder e' ,
e = λ e N C + ( 1 − λ ) e ′ . {\displaystyle e=\lambda e^{NC}+(1-\lambda )e'\,.}
The maximum value of λ over all such decompositions is the noncontextual fraction of e denoted NCF( e ), while the remainder CF( e )=(1-NCF( e )) is the contextual fraction of e . The idea is that we look for a noncontextual explanation for the highest possible fraction of the data, and what is left over is the irreducibly contextual part. Indeed, for any such decomposition that maximises λ the leftover e' is known to be strongly contextual. This measure of contextuality takes values in the interval [0,1], where 0 corresponds to noncontextuality and 1 corresponds to strong contextuality. The contextual fraction may be computed using linear programming .
It has also been proved that CF( e ) is an upper bound on the extent to which e violates any normalised noncontextuality inequality. [ 11 ] Here normalisation means that violations are expressed as fractions of the algebraic maximum violation of the inequality. Moreover, the dual linear program to that which maximises λ computes a noncontextual inequality for which this violation is attained. In this sense the contextual fraction is a more neutral measure of contextuality, since it optimises over all possible noncontextual inequalities rather than checking the statistics against one inequality in particular.
Several measures of the degree of contextuality in contextual systems were proposed within the CbD framework, [ 22 ] but only one of them, denoted CNT 2 , has been shown to naturally extend into a measure of noncontextuality in noncontextual systems, NCNT 2 . This is important, because at least in the non-physical applications of CbD contextuality and noncontextuality are of equal interest. Both CNT 2 and NCNT 2 are defined as the L 1 {\displaystyle L_{1}} -distance between a probability vector p {\displaystyle \mathbf {p} } representing a system and the surface of the noncontextuality polytope P {\displaystyle \mathbb {P} } representing all possible noncontextual systems with the same single-variable marginals. For cyclic systems of dichotomous random variables, it is shown [ 42 ] that if the system is contextual (i.e., D ( C n ) > Δ ( C n ) {\displaystyle D\left({\mathcal {C}}_{n}\right)>\Delta \left({\mathcal {C}}_{n}\right)} ),
C N T 2 = D ( C n ) − Δ ( C n ) , {\displaystyle \mathrm {CNT} _{2}=D\left({\mathcal {C}}_{n}\right)-\Delta \left({\mathcal {C}}_{n}\right),}
and if it is noncontextual ( D ( C n ) ≤ Δ ( C n ) {\displaystyle D\left({\mathcal {C}}_{n}\right)\leq \Delta \left({\mathcal {C}}_{n}\right)} ),
N C N T 2 = min ( Δ ( C n ) − D ( C n ) , m ( C n ) ) , {\displaystyle \mathrm {NCNT} _{2}=\min \left(\Delta \left({\mathcal {C}}_{n}\right)-D\left({\mathcal {C}}_{n}\right),m\left({\mathcal {C}}_{n}\right)\right),}
where m ( C n ) {\displaystyle m\left({\mathcal {C}}_{n}\right)} is the L 1 {\displaystyle L_{1}} -distance from the vector p ∈ P {\displaystyle \mathbf {p} \in \mathbb {P} } to the surface of the box circumscribing the noncontextuality polytope. More generally, NCNT 2 and CNT 2 are computed by means of linear programming. [ 22 ] The same is true for other CbD-based measures of contextuality. One of them, denoted CNT 3 , uses the notion of a quasi-coupling , that differs from a coupling in that the probabilities in the joint distribution of its values are replaced with arbitrary reals (allowed to be negative but summing to 1). The class of quasi-couplings S {\displaystyle S} maximizing the probabilities Pr [ S q c = S q c ′ ] {\displaystyle \Pr \left[S_{q}^{c}=S_{q}^{c'}\right]} is always nonempty, and the minimal total variation of the signed measure in this class is a natural measure of contextuality. [ 43 ]
Recently, quantum contextuality has been investigated as a source of quantum advantage and computational speedups in quantum computing .
Magic state distillation is a scheme for quantum computing in which quantum circuits constructed only of Clifford operators, which by themselves are fault-tolerant but efficiently classically simulable, are injected with certain "magic" states that promote the computational power to universal fault-tolerant quantum computing. [ 44 ] In 2014, Mark Howard, et al. showed that contextuality characterizes magic states for qubits of odd prime dimension and for qubits with real wavefunctions. [ 45 ] Extensions to the qubit case have been investigated by Juani Bermejo Vega et al. [ 41 ] This line of research builds on earlier work by Ernesto Galvão, [ 40 ] which showed that Wigner function negativity is necessary for a state to be "magic"; it later emerged that Wigner negativity and contextuality are in a sense equivalent notions of nonclassicality. [ 46 ]
Measurement-based quantum computation (MBQC) is a model for quantum computing in which a classical control computer interacts with a quantum system by specifying measurements to be performed and receiving measurement outcomes in return. The measurement statistics for the quantum system may or may not exhibit contextuality. A variety of results have shown that the presence of contextuality enhances the computational power of an MBQC.
In particular, researchers have considered an artificial situation in which the power of the classical control computer is restricted to only being able to compute linear Boolean functions, i.e. to solve problems in the Parity L complexity class ⊕ L . For interactions with multi-qubit quantum systems a natural assumption is that each step of the interaction consists of a binary choice of measurement which in turn returns a binary outcome. An MBQC of this restricted kind is known as an l2 -MBQC. [ 47 ]
In 2009, Janet Anders and Dan Browne showed that two specific examples of nonlocality and contextuality were sufficient to compute a non-linear function. This in turn could be used to boost computational power to that of a universal classical computer, i.e. to solve problems in the complexity class P . [ 48 ] This is sometimes referred to as measurement-based classical computation. [ 49 ] The specific examples made use of the Greenberger–Horne–Zeilinger nonlocality proof and the supra-quantum Popescu–Rohrlich box.
In 2013, Robert Raussendorf showed more generally that access to strongly contextual measurement statistics is necessary and sufficient for an l2 -MBQC to compute a non-linear function. He also showed that to compute non-linear Boolean functions with sufficiently high probability requires contextuality. [ 47 ]
A further generalization and refinement of these results due to Samson Abramsky, Rui Soares Barbosa and Shane Mansfield appeared in 2017, proving a precise quantifiable relationship between the probability of successfully computing any given non-linear function and the degree of contextuality present in the l2 -MBQC as measured by the contextual fraction. [ 11 ] Specifically, ( 1 − p s ) ≥ ( 1 − C F ( e ) ) . ν ( f ) {\displaystyle (1-p_{s})\geq \left(1-CF(e)\right).\nu (f)} where p s , C F ( e ) , ν ( f ) ∈ [ 0 , 1 ] {\displaystyle p_{s},CF(e),\nu (f)\in [0,1]} are the probability of success, the contextual fraction of the measurement statistics e , and a measure of the non-linearity of the function to be computed f {\displaystyle f} , respectively. | https://en.wikipedia.org/wiki/Quantum_contextuality |
Quantum coupling is an effect in quantum mechanics in which two or more quantum systems are bound such that a change in one of the quantum states in one of the systems will cause an instantaneous change in all of the bound systems. It is a state similar to quantum entanglement , but whereas quantum entanglement can take place over long distances, quantum coupling is restricted to quantum scales.
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_coupling |
A quantum critical point is a point in the phase diagram of a material where a continuous phase transition takes place at absolute zero . A quantum critical point is typically achieved by a continuous suppression of a nonzero temperature phase transition to zero temperature by the application of a pressure, field, or through doping. Conventional phase transitions occur at nonzero temperature when the growth of random thermal fluctuations leads to a change in the physical state of a system. Condensed matter physics research over the past few decades has revealed a new class of phase transitions called quantum phase transitions [ 1 ] which take place at absolute zero . In the absence of the thermal fluctuations which trigger conventional phase transitions, quantum phase transitions are driven by the zero point quantum fluctuations associated with Heisenberg's uncertainty principle .
Within the class of phase transitions , there are two main categories: at a first-order phase transition , the properties shift discontinuously, as in the melting of solid, whereas at a second order phase transition , the state of the system changes in a continuous fashion. Second-order phase transitions are marked by the growth of fluctuations on ever-longer length-scales. These fluctuations are called "critical fluctuations". At the critical point where a second-order transition occurs the critical fluctuations are scale invariant and extend over the entire system. At a nonzero temperature phase transition, the fluctuations that develop at a critical point are governed by classical physics, because the characteristic energy of quantum fluctuations is always smaller than the characteristic Boltzmann thermal energy k B T {\displaystyle k_{B}T} .
At a quantum critical point, the critical fluctuations are quantum mechanical in nature, exhibiting scale invariance in both space and in time. Unlike classical critical points, where the critical fluctuations are limited to a narrow region around the phase transition, the influence of a quantum critical point is felt over a wide range of temperatures above the quantum critical point, so the effect of quantum criticality is felt without ever reaching absolute zero. Quantum criticality was first observed in ferroelectrics , in which the ferroelectric transition temperature is suppressed to zero.
A wide variety of metallic ferromagnets and antiferromagnets have been observed to develop quantum critical behavior when their magnetic transition temperature is driven to zero through the application of pressure, chemical doping or magnetic fields. In these cases, the properties of the metal are radically transformed by the critical fluctuations, departing qualitatively from the standard Fermi liquid behavior, to form a metallic state sometimes called a non-Fermi liquid or a "strange metal". There is particular interest in these unusual metallic states, which are believed to exhibit a marked preponderance towards the development of superconductivity . Quantum critical fluctuations have also been shown to drive the formation of exotic magnetic phases in the vicinity of quantum critical points. [ 2 ]
Quantum critical points arise when a susceptibility diverges at zero temperature. There are a number of materials (such as CeNi 2 Ge 2 [ 3 ] ) where this occurs serendipitously. More frequently a material has to be tuned to a quantum critical point. Most commonly this is done by taking a system with a second-order phase transition which occurs at nonzero temperature and tuning it—for example by applying pressure or magnetic field or changing its chemical composition. CePd 2 Si 2 is such an example, [ 4 ] where the antiferromagnetic transition which occurs at about 10K under ambient pressure can be tuned to zero temperature by applying a pressure of 28,000 atmospheres. [ 5 ] Less commonly a first-order transition can be made quantum critical. First-order transitions do not normally show critical fluctuations as the material moves discontinuously from one phase into another. However, if the first order phase transition does not involve a change of symmetry then the phase diagram can contain a critical endpoint where the first-order phase transition terminates. Such an endpoint has a divergent susceptibility. The transition between the liquid and gas phases is an example of a first-order transition without a change of symmetry and the critical endpoint is characterized by critical fluctuations known as critical opalescence .
A quantum critical endpoint arises when a nonzero temperature critical point is tuned to zero temperature. One of the best studied examples occurs in the layered ruthenate metal, Sr 3 Ru 2 O 7 in a magnetic field. [ 6 ] This material shows metamagnetism with a low-temperature first-order metamagnetic transition where the magnetization jumps when a magnetic field is applied within the directions of the layers. The first-order jump terminates in a critical endpoint at about 1 kelvin. By switching the direction of the magnetic field so that it points almost perpendicular to the layers, the critical endpoint is tuned to zero temperature at a field of about 8 teslas. The resulting quantum critical fluctuations dominate the physical properties of this material at nonzero temperatures and away from the critical field. The resistivity shows a non-Fermi liquid response, the effective mass of the electron grows and the magnetothermal expansion of the material is modified all in response to the quantum critical fluctuations. | https://en.wikipedia.org/wiki/Quantum_critical_point |
Quantum crystallography is a branch of crystallography that investigates crystalline materials within the framework of quantum mechanics , with analysis and representation, in position or in momentum space , of quantities like wave function , electron charge and spin density , density matrices and all properties related to them (like electric potential, electric or magnetic moments, energy densities, electron localization function, one electron potential, etc.).
Like the quantum chemistry , Quantum crystallography involves both experimental and computational work. The theoretical part of quantum crystallography is based on quantum mechanical calculations of atomic/molecular/crystal wave functions, density matrices or density models, used to simulate the electronic structure of a crystalline material. While in quantum chemistry , the experimental works mainly rely on spectroscopy, in quantum crystallography the scattering techniques ( X-rays , neutrons , γ-Rays , electrons ) play the central role, although spectroscopy as well as atomic microscopy are also sources of information.
The connection between crystallography and quantum chemistry has always been very tight, [ 1 ] after X-ray diffraction techniques became available in crystallography. In fact, the scattering of radiation enables mapping the one-electron distribution [ 2 ] [ 3 ] [ 4 ] or the elements of a density matrix. [ 5 ] The kind of radiation and scattering determines the quantity which is represented (electron charge or spin) and the space in which it is represented (position or momentum space).
Although the wave function is typically assumed not to be directly measurable, recent advances enable also to compute wave functions that are restrained to some experimentally measurable observable (like the scattering of a radiation). [ 6 ] [ 7 ]
The term Quantum Crystallography was first introduced in revisitation articles by L. Huang, L. Massa and Nobel Prize winner Jerome Karle , [ 8 ] [ 9 ] who associated it with two mainstreams: a) crystallographic information that enhances quantum mechanical calculations and b) quantum mechanical approaches to improve crystallography information. This definition mainly refers to studies started in the 1960s and 1970s, when first attempts to obtain wave functions from scattering experiments appeared, [ 10 ] together with other methods to constrain a wavefunction to experimental observations like the dipole moment. [ 11 ] [ 12 ] This field has been recently reviewed, within the context of this definition. [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ]
Parallel to studies on wave function determination, R. F. Stewart [ 18 ] and P. Coppens [ 19 ] [ 20 ] investigated the possibilities to compute models for one-electron charge density from X-ray scattering (for example by means of pseudoatoms multipolar expansion ), and later of spin density from polarized neutron diffraction , [ 21 ] that originated the scientific community of charge, spin and momentum density. [ 22 ] In a recent review article, V. Tsirelson [ 23 ] gave a more general definition: "Quantum crystallography is a research area exploiting the fact that parameters of quantum-mechanically valid electronic model of a crystal can be derived from the accurately measured set of X-ray coherent diffraction structure factors".
The book Modern Charge Density Analysis offers a survey of the research involving Quantum Crystallography and of the most adopted experimental or theoretical methodologies. [ 24 ]
The International Union of Crystallography has recently established a commission on Quantum Crystallography, as extension of the previous commission on Charge, Spin and Momentum density, with the purpose of coordinating research activities in this field. [ 25 ]
The Erice School of crystallography (52nd course): first course on Quantum crystallography (June 2018) The XIX Sagamore Conference (July 2018) The CECAM meeting on Quantum crystallography (June 2017) The IUCr commission on Quantum crystallography The International Union of Crystallography | https://en.wikipedia.org/wiki/Quantum_crystallography |
A quantum depolarizing channel is a model for quantum noise in quantum systems. The d {\displaystyle d} -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map Δ λ {\displaystyle \Delta _{\lambda }} , depending on one parameter λ {\displaystyle \lambda } , which maps a state ρ {\displaystyle \rho } onto a linear combination of itself and the maximally mixed state ,
The condition of complete positivity requires λ {\displaystyle \lambda } to satisfy the bounds
The single qubit depolarizing channel has operator-sum representation [ 1 ] on a density matrix ρ {\displaystyle \rho } given by
where K i {\displaystyle K_{i}} are the Kraus operators given by
and { I , X , Y , Z } {\displaystyle \{I,X,Y,Z\}} are the Pauli matrices . The trace preserving condition is satisfied by the fact that ∑ i K i † K i = I . {\displaystyle \sum _{i}K_{i}^{\dagger }K_{i}=I.}
Geometrically the depolarizing channel Δ λ {\displaystyle \Delta _{\lambda }} can be interpreted as a uniform contraction of the Bloch sphere , parameterized by λ {\displaystyle \lambda } . In the case where λ = 1 {\displaystyle \lambda =1} the channel returns the maximally-mixed state for any input state ρ {\displaystyle \rho } , which corresponds of the complete contraction of the Bloch-sphere down to the single-point I 2 {\displaystyle {\frac {I}{2}}} given by the origin.
The HSW theorem states that the classical capacity of a quantum channel Ψ {\displaystyle \Psi } can be characterized as its regularized Holevo information :
This quantity is difficult to compute and this reflects our ignorance on quantum channels. However, if the Holevo information is additive for a channel Ψ {\displaystyle \Psi } , i.e.,
Then we can get its classical capacity by computing the Holevo information of the channel.
The additivity of Holevo information for all channels was a famous open conjecture in quantum information theory, but it is now known that this conjecture doesn't hold in general. This was proved by showing that the additivity of minimum output entropy for all channels doesn't hold, [ 2 ] which is an equivalent conjecture.
Nonetheless, the additivity of the Holevo information is shown to hold for the quantum depolarizing channel, [ 3 ] and an outline of the proof is given below. As a consequence, entanglement across multiple uses of the channel cannot increase the classical capacity. In this sense, the channel behaves like a classical channel. To achieve the optimal rate of communication, it suffices to choose an orthonormal basis to encode the message, and perform measurements that project onto to the same basis at the receiving end.
The additivity of Holevo information for the depolarizing channel was proved by Christopher King. [ 3 ] He showed that the maximum output ''p''-norm of the depolarizing channel is multiplicative, which implied the additivity of the minimum output entropy, which is equivalent to the additivity of the Holevo information.
A stronger version of the additivity of the Holevo information is shown for the depolarizing channel Δ λ {\displaystyle \Delta _{\lambda }} . For any channel Ψ , {\displaystyle \Psi ,}
This is implied by the following multiplicativity of maximum output p -norm (denoted as v p {\displaystyle v_{p}} ):
The greater than or equal to direction of the above is trivial, it suffices to take the tensor product the states that achieve the maximum p -norm for Δ λ {\displaystyle \Delta _{\lambda }} and Ψ {\displaystyle \Psi } respectively, and input the product state into the product channel to get the output p -norm v p ( Δ λ ) v p ( Ψ ) {\displaystyle v_{p}(\Delta _{\lambda })v_{p}(\Psi )} . The proof for the other direction is more involved
The main idea of the proof is to rewrite the depolarizing channel as a convex combination of simpler channels, and use properties of those simpler channels to get the multiplicativity of the maximum output p -norm for the depolarizing channel.
It turns out that we can write the depolarizing channel as follows:
where c n {\displaystyle c_{n}} 's are positive numbers, U n {\displaystyle U_{n}} 's are unitary matrices, Φ λ ( n ) {\displaystyle \Phi _{\lambda }^{(n)}} 's are some dephasing channels and ρ {\displaystyle \rho } is an arbitrary input state.
Therefore, the product channel can be written as
By the convexity and the unitary invariance of the p -norm, it suffices to show the simpler bound
One important mathematical tool used in the proof of this bound is the Lieb–Thirring inequality , which provides a bound for p -norm of a product of positive matrices. The details and the calculations of the proof are skipped, interested readers are referred to the paper of C. King mentioned above.
The main technique used in this proof, namely rewriting the channel of interest as a convex combination of other simpler channels, is a generalization of the method used earlier to prove similar results for unital qubit channels . [ 4 ]
The fact that the classical capacity of the depolarizing channel is equal to the Holevo information of the channel means that we can't really use quantum effects such as entanglement to improve the transmission rate of classical information. In this sense, the depolarizing channel can be treated as a classical channel.
However the fact that the additivity of Holevo information doesn't hold in general proposes some areas of future work, namely finding channels that violates the additivity, in other words, channels that can exploit quantum effects to improve the classical capacity beyond its Holevo information. | https://en.wikipedia.org/wiki/Quantum_depolarizing_channel |
In condensed matter physics , the quantum dimer magnet state is one in which quantum spins in a magnetic structure entangle to form a singlet state . These entangled spins act as bosons and their excited states (triplons) can undergo Bose-Einstein condensation (BEC). [ 1 ] [ 2 ] The quantum dimer system was originally proposed by Matsubara and Matsuda as a mapping of the lattice Bose gas to the quantum antiferromagnet . [ 3 ] Quantum dimer magnets are often confused as valence bond solids ; however, a valence bond solid requires the breaking of translational symmetry and the dimerizing of spins. In contrast, quantum dimer magnets exist in crystal structures where the translational symmetry is inherently broken. There are two types of quantum dimer models: the XXZ model and the weakly-coupled dimer model. The main difference is the regime in which BEC can occur. For the XXZ model (commonly referred to as the magnon BEC), the BEC occurs upon cooling without a magnetic field and manifests itself as a symmetric dome in the field versus temperature phase diagram centered about H = 0. The weakly-coupled dimer model does not magnetically order in zero magnetic field, but instead orders upon the closing of the spin gap, where the BEC regime begins and is a dome centered at non-zero field.
Quantum dimer systems are considered to be of interest due to their relatively simple interactions and their BEC state is of interest as a novel playground for testing BEC physics. In addition, the BEC state of the quantum dimer magnet is thought to be a spin superfluid which could allow for the transfer of spin information over long distances without loss. [ 4 ]
The Bose-Einstein condensation in quantum dimer systems is, at its essence, a field-induced magnetically ordered state that comes about from the Zeeman splitting of the triplet states. The bosons of the Bose-Einstein condensate can be thought of as the component of the spin parallel to the applied magnetic field, reaching a maximum when the spins become polarized by the field. The difference between the Bose-Einstein condensation and a typical ordered state is the spontaneous breaking of the spin's U(1) symmetry (i.e. the circular symmetry transverse to an applied magnetic field). This spontaneous symmetry breaking gives rise to a Goldstone boson that is measureable via a inelastic neutron scattering (amongst other techniques). [ 5 ] | https://en.wikipedia.org/wiki/Quantum_dimer_magnet |
Quantum dimer models were introduced to model the physics of resonating valence bond (RVB) states in lattice spin systems . The only degrees of freedom retained from the motivating spin systems are the valence bonds, represented as dimers which live on the lattice bonds. In typical dimer models, the dimers do not overlap ("hardcore constraint").
Typical phases of quantum dimer models tend to be valence bond crystals . However, on non-bipartite lattices, RVB liquid phases possessing topological order and fractionalized spinons also appear. The discovery of topological order in quantum dimer models (more than a decade after the models were introduced) has led to new interest in these models.
Classical dimer models have been studied previously in statistical physics , in particular by P. W. Kasteleyn (1961) and
M. E. Fisher (1961).
Exact solution for classical dimer models on planar graphs:
Introduction of model; early literature:
Topological order in quantum dimer model on non-bipartite lattices:
Topological order in quantum spin model on non-bipartite lattices:
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_dimer_models |
Quantum dissipation is the branch of physics that studies the quantum analogues of the process of irreversible loss of energy observed at the classical level. Its main purpose is to derive the laws of classical dissipation from the framework of quantum mechanics . It shares many features with the subjects of quantum decoherence and quantum theory of measurement .
The typical approach to describe dissipation is to split the total system in two parts: the quantum system where dissipation occurs, and a so-called environment or bath into which the energy of the former will flow. The way both systems are coupled depends on the details of the microscopic model, and hence, the description of the bath. To include an irreversible flow of energy (i.e., to avoid Poincaré recurrences in which the energy eventually flows back to the system), requires that the bath contain an infinite number of degrees of freedom. Notice that by virtue of the principle of universality , it is expected that the particular description of the bath will not affect the essential features of the dissipative process, as far as the model contains the minimal ingredients to provide the effect.
The simplest way to model the bath was proposed by Feynman and Vernon in a seminal paper from 1963. [ 1 ] In this description the bath is a sum of an infinite number of harmonic oscillators, that in quantum mechanics represents a set of free bosonic particles.
In 1981, Amir Caldeira and Anthony J. Leggett proposed a simple model to study in detail the way dissipation arises from a quantum point of view. [ 2 ] It describes a quantum particle in one dimension coupled to a bath. The Hamiltonian reads:
The first two terms correspond to the Hamiltonian of a quantum particle of mass M {\displaystyle M} and momentum P {\displaystyle P} , in a potential V {\displaystyle V} at position X {\displaystyle X} . The third term describes the bath as an infinite sum of harmonic oscillators with masses m i {\displaystyle m_{i}} and momentum p i {\displaystyle p_{i}} , at positions q i {\displaystyle q_{i}} . ω i {\displaystyle \omega _{i}} are the frequencies of the harmonic oscillators. The next term describes the way that the system and bath are coupled. In the Caldeira–Leggett model, the bath is coupled to the position of the particle. C i {\displaystyle C_{i}} are coefficients which depend on the details of the coupling. The last term is a counter-term which must be included to ensure that dissipation is homogeneous in all space. As the bath couples to the position, if this term is not included the model is not translationally invariant , in the sense that the coupling is different wherever the quantum particle is located. This gives rise to an unphysical renormalization of the potential, which can be shown to be suppressed by employing real potentials. [ 3 ]
To provide a good description of the dissipation mechanism, a relevant quantity is the bath spectral function, defined as follows:
The bath spectral function provides a constraint in the choice of the coefficients C i {\displaystyle C_{i}} . When this function has the form J ( ω ) = η ω {\displaystyle J(\omega )=\eta \omega } , [ clarification needed ] the corresponding classical kind of dissipation can be shown to be Ohmic . A more generic form is J ( ω ) ∝ ω s {\displaystyle J(\omega )\propto \omega ^{s}} . In this case, if s > 1 {\displaystyle s>1} the dissipation is called "super-ohmic", while if s < 1 {\displaystyle s<1} is sub-ohmic. An example of a super-ohmic bath is the electro-magnetic field under certain circumstances.
As mentioned, the main idea in the field of quantum dissipation is to explain the way classical dissipation can be described from a quantum mechanics point of view. To get the classical limit of the Caldeira–Leggett model, the bath must be integrated out (or traced out ), which can be understood as taking the average over all the possible realizations of the bath and studying the effective dynamics of the quantum system. As a second step, the limit ℏ → 0 {\displaystyle \hbar \rightarrow 0} must be taken to recover classical mechanics . To proceed with those technical steps mathematically, the path integral description of quantum mechanics is usually employed. The resulting classical equations of motion are:
where:
is a kernel which characterizes the effective force that affects the motion of the particle in the presence of dissipation. For so-called Markovian baths , which do not keep memory of the interaction with the system, and for Ohmic dissipation, the equations of motion simplify to the classical equations of motion of a particle with friction:
Hence, one can see how Caldeira–Leggett model fulfills the goal of getting classical dissipation from the quantum mechanics framework. The Caldeira–Leggett model has been used to study quantum dissipation problems since its introduction in 1981, being extensively used as well in the field of quantum decoherence .
The dissipative two-level system is a particular realization of the Caldeira–Leggett model that deserves special attention due to its interest in the field of quantum computation . The aim of the model is to study the effects of dissipation in the dynamics of a particle that can hop between two different positions rather than a continuous degree of freedom. This reduced Hilbert space allows the problem to be described in terms of 1 / 2 - spin operators. This is sometimes referred in the literature as the spin-boson model, and it is closely related to the Jaynes–Cummings model .
The Hamiltonian for the dissipative two-level system reads:
H = Δ σ x 2 + ∑ i ( p i 2 2 m i + 1 2 m i ω i 2 q i 2 ) + σ z 2 ∑ i C i q i {\displaystyle H=\Delta {\frac {\sigma _{x}}{2}}+\sum _{i}\left({\frac {p_{i}^{2}}{2m_{i}}}+{\frac {1}{2}}m_{i}\omega _{i}^{2}q_{i}^{2}\right)+{\frac {\sigma _{z}}{2}}\sum _{i}{C_{i}q_{i}}} ,
where σ x {\displaystyle \sigma _{x}} and σ z {\displaystyle \sigma _{z}} are the Pauli matrices and Δ {\displaystyle \Delta } is the amplitude of hopping between the two possible positions. Notice that in this model the counter-term is no longer needed, as the coupling to S z {\displaystyle S_{z}} gives already homogeneous dissipation.
The model has many applications. In quantum dissipation, it is used as a simple model to study the dynamics of a dissipative particle confined in a double-well potential. In the context of quantum computation, it represents a qubit coupled to an environment, which can produce decoherence . In the study of amorphous solids , it provides the basis of the standard theory to describe their thermodynamic properties.
The dissipative two-level system represents also a paradigm in the study of quantum phase transitions . For a critical value of the coupling to the bath it shows a phase transition from a regime in which the particle is delocalized among the two positions to another in which it is localized in only one of them. The transition is of Kosterlitz–Thouless kind, as can be seen by deriving the renormalization group flow equations for the hopping term.
A different approach to describe energy dissipation is to consider time dependent Hamiltonians. Against a common misunderstanding, the resulting unitary dynamics can describe energy dissipation, as certain degrees of freedom loose energy and others gain energy. [ 4 ] However, the quantum mechanical state of the system stays pure , thus such an approach can not describe dephasing unless a subsystem is chosen and the reduced density matrix of this open quantum system is analyzed. [ 5 ] Dephasing leads to quantum decoherence or information dissipation and is often important when describing open quantum systems . However, this approach is typically used e.g. in the description of optical experiments. There a light pulse (described by a time dependent semi-classical Hamiltonian) can change the energy in the system by stimulated absorption or emission. [ citation needed ] | https://en.wikipedia.org/wiki/Quantum_dissipation |
Quantum dots ( QDs ) or semiconductor nanocrystals are semiconductor particles a few nanometres in size with optical and electronic properties that differ from those of larger particles via quantum mechanical effects . They are a central topic in nanotechnology and materials science . When a quantum dot is illuminated by UV light , an electron in the quantum dot can be excited to a state of higher energy. In the case of a semiconducting quantum dot, this process corresponds to the transition of an electron from the valence band to the conduction band . The excited electron can drop back into the valence band releasing its energy as light. This light emission ( photoluminescence ) is illustrated in the figure on the right. The color of that light depends on the energy difference between the discrete energy levels of the quantum dot in the conduction band and the valence band .
In other words, a quantum dot can be defined as a structure on a semiconductor which is capable of confining electrons in three dimensions, in this way it is possible to obtain discrete energy levels. Basically, the quantum dots are little crystals but behave as individual atoms, and their properties can be manipulated. [ 1 ]
Nanoscale materials with semiconductor properties tightly confine either electrons or electron holes . The confinement is similar to a three-dimensional particle in a box model. The quantum dot absorption and emission features correspond to transitions between discrete quantum mechanically allowed energy levels in the box that are reminiscent of atomic spectra. For these reasons, quantum dots are sometimes referred to as artificial atoms , [ 2 ] emphasizing their bound and discrete electronic states , like naturally occurring atoms or molecules . [ 3 ] [ 4 ] It was shown that the electronic wave functions in quantum dots resemble the ones in real atoms. [ 5 ]
Quantum dots have properties intermediate between bulk semiconductors and discrete atoms or molecules. Their optoelectronic properties change as a function of both size and shape. [ 6 ] [ 7 ] Larger QDs of 5–6 nm diameter emit longer wavelengths , with colors such as orange, or red. Smaller QDs (2–3 nm) emit shorter wavelengths, yielding colors like blue and green. However, the specific colors vary depending on the exact composition of the QD. [ 8 ]
Potential applications of quantum dots include single-electron transistors , solar cells , LEDs , lasers , [ 9 ] single-photon sources , [ 10 ] [ 11 ] [ 12 ] second-harmonic generation , quantum computing , [ 13 ] cell biology research, [ 14 ] microscopy , [ 15 ] and medical imaging . [ 16 ] Their small size allows for some QDs to be suspended in solution, which may lead to their use in inkjet printing , and spin coating . [ 17 ] They have been used in Langmuir–Blodgett thin films . [ 18 ] [ 19 ] [ 20 ] These processing techniques result in less expensive and less time-consuming methods of semiconductor fabrication .
Quantum dots are usually coated with organic capping ligands (typically with long hydrocarbon chains, such as oleic acid) to control growth, prevent aggregation, and to promote dispersion in solution. [ 21 ] However, these organic coatings can lead to non-radiative recombination after photogeneration, meaning the generated charge carriers can be dissipated without photon emission (e.g. via phonons or trapping in defect states), which reduces fluorescent quantum yield, or the conversion efficiency of absorbed photons into emitted fluorescence. [ 22 ] To combat this, a semiconductor layer can be grown surrounding the quantum dot core. Depending on the bandgaps of the core and shell materials, the fluorescent properties of the nanocrystals can be tuned. Furthermore, adjusting the thicknesses of each of the layers and overall size of the quantum dots can affect the photoluminescent emission wavelength — the quantum confinement effect tends to blueshift the emission spectra as the quantum dot decreases in size. [ 23 ] There are 4 major categories of quantum dot heterostructures: type I, inverse type I, type II, and inverse type II. [ 24 ]
Type I quantum dots are composed of a semiconductor core encapsulated in a second semiconductor material with a larger bandgap, which can passivate non-radiative recombination sites at the surface of the quantum dots and improve quantum yield . Inverse type I quantum dots have a semiconductor layer with a smaller bandgap which leads to delocalized charge carriers in the shell. For type II and inverse type II dots, either the conduction or valence band of the core is located within the bandgap of the shell, which can lead to spatial separation of charge carriers in the core and shell. [ 24 ] For all of these core/shell systems, the deposition of the outer layer can lead to potential lattice mismatch, which can limit the ability to grow a thick shell without reducing photoluminescent performance.
One such reason for the decrease in performance can be attributed to the physical strain being put on the lattice. In a case where ZnSe/ZnS (type I) and ZnSe/CdS (type II) quantum dots were being compared, the diameter of the uncoated ZnSe core (obtained using TEM ) was compared to the capped core diameter (calculated via effective mass approximation model) [lattice strain source] to better understand the effect of core-shell strain. [ 25 ] Type I heterostructures were found to induce compressive strain and “squeeze” the core, while the type II heterostructures had the effect of stretching the core under tensile strain. [ 25 ] Because the fluorescent properties of quantum dots are dictated by nanocrystal size, induced changes in core dimensions can lead to shifting of emission wavelength, further proving why an intermediate semiconductor layer is necessary to rectify lattice mismatch and improve quantum yield. [ 26 ]
One such core/double-shell system is the CdSe/ZnSe/ZnS nanocrystal. [ 26 ] In a study comparing CdSe/ZnS and CdSe/ZnSe nanocrystals, the former was found to have PL yield 84% of the latter’s, due to a lattice mismatch. To study the double-shell system, after synthesis of the core CdSe nanocrystals, a layer of ZnSe was coated prior to the ZnS outer shell, leading to an improvement in fluorescent efficiency by 70%. Furthermore, the two additional layers were found to improve resistance of the nanocrystals against photo-oxidation, which can contribute to degradation of the emission spectra.
It is also standard for surface passivation techniques to be applied to these core/double-shell systems, as well. As mentioned above, oleic acid is one such organic capping ligand that is used to promote colloidal stability and control nanocrystal growth, and can even be used to initiate a second round of ligand exchange and surface functionalization. [ 21 ] [ 27 ] However, because of the detrimental effect organic ligands have on PL efficiency, further studies have been conducted to obtain all-inorganic quantum dots. In one such study, intensely luminescent all-inorganic nanocrystals (ILANs) were synthesized via a ligand exchange process which substituted metal salts for the oleic acid ligands, and were found to have comparable photoluminescent quantum yields to that of existing red- and green-emitting quantum dots. [ 21 ]
There are several ways to fabricate quantum dots. Possible methods include colloidal synthesis, self-assembly , and electrical gating.
Colloidal semiconductor nanocrystals are synthesized from solutions, much like traditional chemical processes . The main difference is the product neither precipitates as a bulk solid nor remains dissolved. [ 6 ] Heating the solution at high temperature, the precursors decompose forming monomers which then nucleate and generate nanocrystals. Temperature is a critical factor in determining optimal conditions for the nanocrystal growth. It must be high enough to allow for rearrangement and annealing of atoms during the synthesis process while being low enough to promote crystal growth. The concentration of monomers is another critical factor that has to be stringently controlled during nanocrystal growth. The growth process of nanocrystals can occur in two different regimes: "focusing" and "defocusing". At high monomer concentrations, the critical size (the size where nanocrystals neither grow nor shrink) is relatively small, resulting in growth of nearly all particles. In this regime, smaller particles grow faster than large ones (since larger crystals need more atoms to grow than small crystals) resulting in the size distribution focusing , yielding an improbable distribution of nearly monodispersed particles. The size focusing is optimal when the monomer concentration is kept such that the average nanocrystal size present is always slightly larger than the critical size. Over time, the monomer concentration diminishes, the critical size becomes larger than the average size present, and the distribution defocuses .
There are colloidal methods to produce many different semiconductors. Typical dots are made of binary compounds such as lead sulfide , lead selenide , cadmium selenide , cadmium sulfide , cadmium telluride , indium arsenide , and indium phosphide . Dots may also be made from ternary compounds such as cadmium selenide sulfide. Further, recent advances have been made which allow for synthesis of colloidal perovskite quantum dots. [ 28 ] These quantum dots can contain as few as 100 to 100,000 atoms within the quantum dot volume, with a diameter of approximately 10 to 50 atom diameters. This corresponds to about 2 to 10 nanometers , and at 10 nm in diameter, nearly 3 million quantum dots could be lined up end to end and fit within the width of a human thumb.
Large batches of quantum dots may be synthesized via colloidal synthesis . Due to this scalability and the convenience of benchtop conditions , colloidal synthetic methods are promising for commercial applications.
Plasma synthesis has evolved to be one of the most popular gas-phase approaches for the production of quantum dots, especially those with covalent bonds. [ 29 ] [ 30 ] [ 31 ] For example, silicon and germanium quantum dots have been synthesized by using nonthermal plasma . The size, shape, surface and composition of quantum dots can all be controlled in nonthermal plasma. [ 32 ] [ 33 ] Doping that seems quite challenging for quantum dots has also been realized in plasma synthesis. [ 34 ] [ 35 ] [ 36 ] Quantum dots synthesized by plasma are usually in the form of powder, for which surface modification may be carried out. This can lead to excellent dispersion of quantum dots in either organic solvents [ 37 ] or water [ 38 ] (i. e., colloidal quantum dots).
The electrostatic potential needed to create a quantum dot can be realized with several methods. These include external electrodes, [ 39 ] doping, strain, [ 40 ] or impurities. Self-assembled quantum dots are typically between 5 and 50 nm in size. Quantum dots defined by lithographically patterned gate electrodes, or by etching on two-dimensional electron gases in semiconductor heterostructures can have lateral dimensions between 20 and 100 nm.
The formation of quantum dots can be spontaneous when a semiconductor material is deposited on a substrate and a difference in lattice space exists between them. [ 1 ]
By means of advanced nanofabrication technologies it is possible to manipulate properties of the quantum dots, such as their interactions, shape, size and transparency.
For example, when negative voltage is applied to a metal gate around a QD, as response, its diameter starts to be gradually squeezed, as a consequence, the number of electrons on the dot starts to decrease one by one, this could be made until there are no more left.
The previous property allows to record the current flow as the number of electrons on the dot, this implies that the energy variates. [ 41 ]
Genetically engineered M13 bacteriophage viruses allow preparation of quantum dot biocomposite structures. [ 47 ] It had previously been shown that genetically engineered viruses can recognize specific semiconductor surfaces through the method of selection by combinatorial phage display . [ 48 ] Additionally, it is known that liquid crystalline structures of wild-type viruses (Fd, M13, and TMV ) are adjustable by controlling the solution concentrations, solution ionic strength , and the external magnetic field applied to the solutions. Consequently, the specific recognition properties of the virus can be used to organize inorganic nanocrystals, forming ordered arrays over the length scale defined by liquid crystal formation. Using this information, Lee et al. (2000) [ citation needed ] were able to create self-assembled, highly oriented, self-supporting films from a phage and ZnS precursor solution. This system allowed them to vary both the length of bacteriophage and the type of inorganic material through genetic modification and selection.
Highly ordered arrays of quantum dots may also be self-assembled by electrochemical techniques. A template is created by causing an ionic reaction at an electrolyte–metal interface which results in the spontaneous assembly of nanostructures, including quantum dots, onto the metal which is then used as a mask for mesa-etching these nanostructures on a chosen substrate. [ citation needed ]
Quantum dot manufacturing relies on a process called high temperature dual injection which has been scaled by multiple companies for commercial applications that require large quantities (hundreds of kilograms to tons) of quantum dots. This reproducible production method can be applied to a wide range of quantum dot sizes and compositions.
The bonding in certain cadmium-free quantum dots, such as III–V -based quantum dots, is more covalent than that in II–VI materials, therefore it is more difficult to separate nanoparticle nucleation and growth via a high temperature dual injection synthesis. An alternative method of quantum dot synthesis, the molecular seeding process, provides a reproducible route to the production of high-quality quantum dots in large volumes. The process utilises identical molecules of a molecular cluster compound as the nucleation sites for nanoparticle growth, thus avoiding the need for a high temperature injection step. Particle growth is maintained by the periodic addition of precursors at moderate temperatures until the desired particle size is reached. [ 49 ] The molecular seeding process is not limited to the production of cadmium-free quantum dots; for example, the process can be used to synthesise kilogram batches of high-quality II–VI quantum dots in just a few hours.
Another approach for the mass production of colloidal quantum dots can be seen in the transfer of the well-known hot-injection methodology for the synthesis to a technical continuous flow system. The batch-to-batch variations arising from the needs during the mentioned methodology can be overcome by utilizing technical components for mixing and growth as well as transport and temperature adjustments. For the production of CdSe based semiconductor nanoparticles this method has been investigated and tuned to production amounts of kilograms per month. Since the use of technical components allows for easy interchange in regards of maximum throughput and size, it can be further enhanced to tens or even hundreds of kilograms. [ 50 ]
In 2011 a consortium of U.S. and Dutch companies reported a milestone in high-volume quantum dot manufacturing by applying the traditional high temperature dual injection method to a flow system . [ 51 ]
On 23 January 2013 Dow entered into an exclusive licensing agreement with UK-based Nanoco for the use of their low-temperature molecular seeding method for bulk manufacture of cadmium-free quantum dots for electronic displays, and on 24 September 2014 Dow commenced work on the production facility in South Korea capable of producing sufficient quantum dots for "millions of cadmium-free televisions and other devices, such as tablets". Mass production was due to commence in mid-2015. [ 52 ] On 24 March 2015, Dow announced a partnership deal with LG Electronics to develop the use of cadmium free quantum dots in displays. [ 53 ]
Some quantum dots pose risks to human health and the environment under certain conditions. [ 54 ] [ 55 ] [ 56 ] Notably, the studies on quantum dot toxicity have focused on particles containing cadmium and have yet to be demonstrated in animal models after physiologically relevant dosing. [ 56 ] In vitro studies, based on cell cultures, on quantum dots (QD) toxicity suggest that their toxicity may derive from multiple factors including their physicochemical characteristics (size, shape, composition, surface functional groups, and surface charges) and their environment. Assessing their potential toxicity is complex as these factors include properties such as QD size, charge, concentration, chemical composition, capping ligands, and also on their oxidative, mechanical, and photolytic stability. [ 54 ]
Many studies have focused on the mechanism of QD cytotoxicity using model cell cultures. It has been demonstrated that after exposure to ultraviolet radiation or oxidation by air, CdSe QDs release free cadmium ions causing cell death. [ 57 ] Group II–VI QDs also have been reported to induce the formation of reactive oxygen species after exposure to light, which in turn can damage cellular components such as proteins, lipids, and DNA. [ 58 ] Some studies have also demonstrated that addition of a ZnS shell inhibits the process of reactive oxygen species in CdSe QDs. Another aspect of QD toxicity is that there are, in vivo, size-dependent intracellular pathways that concentrate these particles in cellular organelles that are inaccessible by metal ions, which may result in unique patterns of cytotoxicity compared to their constituent metal ions. [ 59 ] The reports of QD localization in the cell nucleus [ 60 ] present additional modes of toxicity because they may induce DNA mutation, which in turn will propagate through future generation of cells, causing diseases.
Although concentration of QDs in certain organelles have been reported in in vivo studies using animal models, no alterations in animal behavior, weight, hematological markers, or organ damage has been found through either histological or biochemical analysis. [ 61 ] These findings have led scientists to believe that intracellular dose is the most important determining factor for QD toxicity. Therefore, factors determining the QD endocytosis that determine the effective intracellular concentration, such as QD size, shape, and surface chemistry determine their toxicity. Excretion of QDs through urine in animal models also have demonstrated via injecting radio-labeled ZnS-capped CdSe QDs where the ligand shell was labeled with 99m Tc . [ 62 ] Though multiple other studies have concluded retention of QDs in cellular levels, [ 56 ] [ 63 ] exocytosis of QDs is still poorly studied in the literature.
While significant research efforts have broadened the understanding of toxicity of QDs, there are large discrepancies in the literature, and questions still remain to be answered. Diversity of this class of material as compared to normal chemical substances makes the assessment of their toxicity very challenging. As their toxicity may also be dynamic depending on the environmental factors such as pH level, light exposure, and cell type, traditional methods of assessing toxicity of chemicals such as LD 50 are not applicable for QDs. Therefore, researchers are focusing on introducing novel approaches and adapting existing methods to include this unique class of materials. [ 56 ] Furthermore, novel strategies to engineer safer QDs are still under exploration by the scientific community. A recent novelty in the field is the discovery of carbon quantum dots , a new generation of optically active nanoparticles potentially capable of replacing semiconductor QDs, but with the advantage of much lower toxicity.
Quantum dots have been gaining interest from the scientific community because of their interesting optical properties, the main being band gap tunability. When an electron is excited to the conduction band, it leaves behind a vacancy in the valence band called hole . These two opposite charges are bound by Coulombic interactions in what is called an exciton and their spatitial separation is defined by the exciton Bohr radius. In a nanostructure of comparable size to the exciton Bohr radius, the exciton is physically confined within the semiconductor resulting in an increase of the band gap of the material. This dependence can be predicted using the Brus model. [ 64 ]
As the confinement energy depends on the quantum dot's size, both absorption onset and fluorescence emission can be tuned by changing the size of the quantum dot during its synthesis. The larger the dot, the redder (lower-energy) its absorption onset and fluorescence spectrum . Conversely, smaller dots absorb and emit bluer (higher-energy) light. Recent articles suggest that the shape of the quantum dot may be a factor in the coloration as well, but as yet not enough information is available [ citation needed ] . Furthermore, it was shown [ 65 ] that the lifetime of fluorescence is determined by the size of the quantum dot. Larger dots have more closely spaced energy levels in which the electron–hole pair can be trapped. Therefore, electron–hole pairs in larger dots live longer causing larger dots to show a longer lifetime.
To improve fluorescence quantum yield , quantum dots can be made with shells of a larger bandgap semiconductor material around them. The improvement is suggested to be due to the reduced access of electron and hole to non-radiative surface recombination pathways in some cases, but also due to reduced Auger recombination in others.
Quantum dots are particularly promising for optical applications due to their high extinction coefficient [ 66 ] and ultrafast optical nonlinearities with potential applications for developing all-optical systems. [ 67 ] They operate like a single-electron transistor and show the Coulomb blockade effect. Quantum dots have also been suggested as implementations of qubits for quantum information processing , [ 68 ] and as active elements for thermoelectrics. [ 69 ] [ 70 ] [ 71 ]
Tuning the size of quantum dots is attractive for many potential applications. For instance, larger quantum dots have a greater spectrum shift toward red compared to smaller dots and exhibit less pronounced quantum properties. Conversely, the smaller particles allow one to take advantage of more subtle quantum effects.
Being zero-dimensional , quantum dots have a sharper density of states than higher-dimensional structures. As a result, they have superior transport and optical properties. They have potential uses in diode lasers , amplifiers, and biological sensors. [ 73 ] Quantum dots may be excited within a locally enhanced electromagnetic field produced by gold nanoparticles, which then can be observed from the surface plasmon resonance in the photoluminescent excitation spectrum of (CdSe)ZnS nanocrystals. High-quality quantum dots are well suited for optical encoding and multiplexing applications due to their broad excitation profiles and narrow/symmetric emission spectra. The new generations of quantum dots have far-reaching potential for the study of intracellular processes at the single-molecule level, high-resolution cellular imaging, long-term in vivo observation of cell trafficking, tumor targeting, and diagnostics.
CdSe nanocrystals are efficient triplet photosensitizers. [ 74 ] Laser excitation of small CdSe nanoparticles enables the extraction of the excited state energy from the quantum dots into bulk solution, thus opening the door to a wide range of potential applications such as photodynamic therapy, photovoltaic devices, molecular electronics, and catalysis.
In modern biological analysis, various kinds of organic dyes are used. However, as technology advances, greater flexibility in these dyes is sought. [ 75 ] To this end, quantum dots have quickly filled in the role, being found to be superior to traditional organic dyes on several counts, one of the most immediately obvious being brightness (owing to the high extinction coefficient combined with a comparable quantum yield to fluorescent dyes [ 14 ] ) as well as their stability (allowing much less photobleaching ). [ 76 ] It has been estimated that quantum dots are 20 times brighter and 100 times more stable than traditional fluorescent reporters. [ 75 ] For single-particle tracking, the irregular blinking of quantum dots is a minor drawback. However, there have been groups which have developed quantum dots which are essentially nonblinking and demonstrated their utility in single-molecule tracking experiments. [ 77 ] [ 78 ]
The use of quantum dots for highly sensitive cellular imaging has seen major advances. [ 79 ] The improved photostability of quantum dots, for example, allows the acquisition of many consecutive focal-plane images that can be reconstructed into a high-resolution three-dimensional image. [ 80 ] Another application that takes advantage of the extraordinary photostability of quantum dot probes is the real-time tracking of molecules and cells over extended periods of time. [ 81 ] Antibodies , streptavidin , [ 82 ] peptides , [ 83 ] DNA , [ 84 ] nucleic acid aptamers , [ 85 ] or small-molecule ligands [ 86 ] can be used to target quantum dots to specific proteins on cells. Researchers were able to observe quantum dots in lymph nodes of mice for more than 4 months. [ 87 ]
Quantum dots can have antibacterial properties similar to nanoparticles and can kill bacteria in a dose-dependent manner. [ 88 ] One mechanism by which quantum dots can kill bacteria is through impairing the functions of antioxidative system in the cells and down regulating the antioxidative genes. In addition, quantum dots can directly damage the cell wall. Quantum dots have been shown to be effective against both gram- positive and gram-negative bacteria. [ 89 ]
Semiconductor quantum dots have also been employed for in vitro imaging of pre-labeled cells. The ability to image single-cell migration in real time is expected to be important to several research areas such as embryogenesis , cancer metastasis , stem cell therapeutics, and lymphocyte immunology .
One application of quantum dots in biology is as donor fluorophores in Förster resonance energy transfer , where the large extinction coefficient and spectral purity of these fluorophores make them superior to molecular fluorophores [ 90 ] It is also worth noting that the broad absorbance of QDs allows selective excitation of the QD donor and a minimum excitation of a dye acceptor in FRET-based studies. [ 91 ] The applicability of the FRET model, which assumes that the Quantum Dot can be approximated as a point dipole, has recently been demonstrated [ 92 ]
The use of quantum dots for tumor targeting under in vivo conditions employ two targeting schemes: active targeting and passive targeting. In the case of active targeting, quantum dots are functionalized with tumor-specific binding sites to selectively bind to tumor cells. Passive targeting uses the enhanced permeation and retention of tumor cells for the delivery of quantum dot probes. Fast-growing tumor cells typically have more permeable membranes than healthy cells, allowing the leakage of small nanoparticles into the cell body. Moreover, tumor cells lack an effective lymphatic drainage system, which leads to subsequent nanoparticle accumulation.
Quantum dot probes exhibit in vivo toxicity. For example, CdSe nanocrystals are highly toxic to cultured cells under UV illumination, because the particles dissolve, in a process known as photolysis , to release toxic cadmium ions into the culture medium. In the absence of UV irradiation, however, quantum dots with a stable polymer coating have been found to be essentially nontoxic. [ 87 ] [ 55 ] Hydrogel encapsulation of quantum dots allows for quantum dots to be introduced into a stable aqueous solution, reducing the possibility of cadmium leakage. Then again, only little is known about the excretion process of quantum dots from living organisms. [ 93 ]
In another potential application, quantum dots are being investigated as the inorganic fluorophore for intra-operative detection of tumors using fluorescence spectroscopy .
Delivery of undamaged quantum dots to the cell cytoplasm has been a challenge with existing techniques. Vector-based methods have resulted in aggregation and endosomal sequestration of quantum dots while electroporation can damage the semi-conducting particles and aggregate delivered dots in the cytosol. Via cell squeezing , quantum dots can be efficiently delivered without inducing aggregation, trapping material in endosomes, or significant loss of cell viability. Moreover, it has shown that individual quantum dots delivered by this approach are detectable in the cell cytosol, thus illustrating the potential of this technique for single-molecule tracking studies. [ 94 ]
The tunable absorption spectrum and high extinction coefficients of quantum dots make them attractive for light harvesting technologies such as photovoltaics. Quantum dots may be able to increase the efficiency and reduce the cost of today's typical silicon photovoltaic cells . According to an experimental report from 2004, [ 95 ] quantum dots of lead selenide (PbSe) can produce more than one exciton from one high-energy photon via the process of carrier multiplication or multiple exciton generation (MEG). This compares favorably to today's photovoltaic cells which can only manage one exciton per high-energy photon, with high kinetic energy carriers losing their energy as heat. On the other hand, the quantum-confined ground-states of colloidal quantum dots (such as lead sulfide , PbS) incorporated in wider-bandgap host semiconductors (such as perovskite ) can allow the generation of photocurrent from photons with energy below the host bandgap, via a two-photon absorption process, offering another approach (termed intermediate band , IB) to exploit a broader range of the solar spectrum and thereby achieve higher photovoltaic efficiency . [ 96 ] [ 97 ]
Colloidal quantum dot photovoltaics would theoretically be cheaper to manufacture, as they can be made using simple chemical reactions.
Aromatic self-assembled monolayers (SAMs) (such as 4-nitrobenzoic acid ) can be used to improve the band alignment at electrodes for better efficiencies. This technique has provided a record power conversion efficiency (PCE) of 10.7%. [ 98 ] The SAM is positioned between ZnO–PbS colloidal quantum dot (CQD) film junction to modify band alignment via the dipole moment of the constituent SAM molecule, and the band tuning may be modified via the density, dipole and the orientation of the SAM molecule. [ 98 ]
Colloidal quantum dots are also used in inorganic–organic hybrid solar cells . These solar cells are attractive because of the potential for low-cost fabrication and relatively high efficiency. [ 99 ] Incorporation of metal oxides, such as ZnO, TiO 2 , and Nb 2 O 5 nanomaterials into organic photovoltaics have been commercialized using full roll-to-roll processing. [ 99 ] A 13.2% power conversion efficiency is claimed in Si nanowire/PEDOT:PSS hybrid solar cells. [ 100 ]
Another potential use involves capped single-crystal ZnO nanowires with CdSe quantum dots, immersed in mercaptopropionic acid as hole transport medium in order to obtain a QD-sensitized solar cell. The morphology of the nanowires allowed the electrons to have a direct pathway to the photoanode. This form of solar cell exhibits 50–60% internal quantum efficiencies . [ 101 ]
Nanowires with quantum dot coatings on silicon nanowires (SiNW) and carbon quantum dots. The use of SiNWs instead of planar silicon enhances the antiflection properties of Si. [ 102 ] The SiNW exhibits a light-trapping effect due to light trapping in the SiNW. This use of SiNWs in conjunction with carbon quantum dots resulted in a solar cell that reached 9.10% PCE. [ 102 ]
Graphene quantum dots have also been blended with organic electronic materials to improve efficiency and lower cost in photovoltaic devices and organic light emitting diodes ( OLEDs ) compared to graphene sheets. These graphene quantum dots were functionalized with organic ligands that experience photoluminescence from UV–visible absorption. [ 103 ]
It can be seen improvements in the electrical conductivity and charge retention of batteries when QDs are added to anodes.
In a comparison made between Pure MnO and MnO doped with quantum dot for the capacity of charge and discharge in (mAh/g) against the number of cycles, it can be seen that battery capacity, or the amount of energy that a battery can hold, is higher in MnO Quantum Dote-doped batteries than in batteries without, and remains higher after many charging/discharging cycles, taking in consideration a current density of Ag^-1. There exists a constant average difference of around 250 mAh/g in favor of the doped compound for both charge and discharge comparisons, comparing from 0 to 60 cycles, going from 1000 mAh/g to 450 mAh/g in the first 60 cycles for the doped compound, and from 750 mAh/g to 200 mAh/g for the pure MnO. [ 104 ]
A comparison using Graphene Quantum Dots for a NP-SiAl compound not only shows higher discharge capacities but also an improved electrochemical impedance spectroscopy plot, indicating that the battery has better electrical conductivity. For the case of the NP-SiAl/GQDs, the value of -Z´´/ohm reaches a peak of 300, for 250 Z´/ohm, while for the pure NP-SiAl, the peak of 300 -Z´´/ohm is reached at 650 Z´/ohm. [ 105 ]
In terms of energy, each individual quantum dot presents an energy level which is compared to that of an atom.Extending this property, an artificial lattice (made out of QDs) would have an energy band structure similar to the one of a crystalline semiconductor.
The energy level of a dot is dependent on the amount of charge in it and its capacitance.The energy present in electrons is proportional to the square of the wavelength, which makes the energy levels to rise quickly [ 106 ]
Carbon quantum dots and Graphite quantum dots are the main types of quantum dots used in batteries.
The graphene quantum dots are made out of graphene sheets which are attached among them, forming a morphology similar to a 2D-disk.
The carbon quantum dots have an isotropic spherical structure and are made out of crystalline and amorphous carbon sheets.
From these common quantum dots, the graphene ones, are usually more crystalline than the carbon ones, this is because they have the crystallinity of a mono-layered and few-layered graphene. [ 105 ]
Several methods are proposed for using quantum dots to improve existing light-emitting diode (LED) design, including quantum dot light-emitting diode (QD-LED or QLED) displays, and quantum dot white-light-emitting diode (QD-WLED) displays. Because quantum dots naturally produce monochromatic light, they can be more efficient than light sources which must be color filtered. QD-LEDs can be fabricated on a silicon substrate, which allows them to be integrated onto standard silicon-based integrated circuits or microelectromechanical systems . [ 107 ]
Quantum dots are valued for displays because they emit light in very specific Gaussian distributions . This can result in a display with visibly more accurate colors.
A conventional color liquid crystal display (LCD) is usually backlit by fluorescent lamps (CCFLs) or conventional white LEDs that are color filtered to produce red, green, and blue pixels. Quantum dot displays use blue-emitting LEDs rather than white LEDs as the light sources. The converting part of the emitted light is converted into pure green and red light by the corresponding color quantum dots placed in front of the blue LED or using a quantum dot infused diffuser sheet in the backlight optical stack. Blank pixels are also used to allow the blue LED light to still generate blue hues. This type of white light as the backlight of an LCD panel allows for the best color gamut at lower cost than an RGB LED combination using three LEDs. [ 108 ]
Another method by which quantum dot displays can be achieved is the electroluminescent (EL) or electro-emissive method. This involves embedding quantum dots in each individual pixel. These are then activated and controlled via an electric current application. [ 109 ] Since this is often light emitting itself, the achievable colors may be limited in this method. [ 110 ] Electro-emissive QD-LED TVs exist in laboratories only.
The ability of QDs to precisely convert and tune a spectrum makes them attractive for LCD displays. Previous LCD displays can waste energy converting red-green poor, blue-yellow rich white light into a more balanced lighting. By using QDs, only the necessary colors for ideal images are contained in the screen. The result is a screen that is brighter, clearer, and more energy-efficient. The first commercial application of quantum dots was the Sony XBR X900A series of flat panel televisions released in 2013. [ 111 ]
In June 2006, QD Vision announced technical success in making a proof-of-concept quantum dot display and show a bright emission in the visible and near infrared region of the spectrum. A QD-LED integrated at a scanning microscopy tip was used to demonstrate fluorescence near-field scanning optical microscopy ( NSOM ) imaging. [ 112 ]
Quantum dot photodetectors (QDPs) can be fabricated either via solution-processing, [ 113 ] or from conventional single-crystalline semiconductors. [ 114 ] Conventional single-crystalline semiconductor QDPs are precluded from integration with flexible organic electronics due to the incompatibility of their growth conditions with the process windows required by organic semiconductors . On the other hand, solution-processed QDPs can be readily integrated with an almost infinite variety of substrates, and also postprocessed atop other integrated circuits. Such colloidal QDPs have potential applications in visible- and infrared -light cameras , [ 115 ] machine vision, industrial inspection, spectroscopy , and fluorescent biomedical imaging.
Quantum dots also function as photocatalysts for the light driven chemical conversion of water into hydrogen as a pathway to solar fuel . In photocatalysis , electron hole pairs formed in the dot under band gap excitation drive redox reactions in the surrounding liquid. Generally, the photocatalytic activity of the dots is related to the particle size and its degree of quantum confinement . [ 116 ] This is because the band gap determines the chemical energy that is stored in the dot in the excited state . An obstacle for the use of quantum dots in photocatalysis is the presence of surfactants on the surface of the dots. These surfactants (or ligands ) interfere with the chemical reactivity of the dots by slowing down mass transfer and electron transfer processes. Also, quantum dots made of metal chalcogenides are chemically unstable under oxidizing conditions and undergo photo corrosion reactions.
Quantum dots can also be used to study fundamental effects in materials science . By coupling two or more such quantum dots, an artificial molecule can be made, exhibiting hybridization even at room temperature. [ 117 ] Precise assembly of quantum dots can form superlattices that act as artificial solid-state materials that exhibit unique optical and electronic properties. [ 118 ] [ 119 ]
Quantum dots are theoretically described as a point-like, or zero dimensional (0D) entity. Most of their properties depend on the dimensions, shape, and materials of which QDs are made. Generally, QDs present different thermodynamic properties from their bulk materials. One of these effects is melting-point depression . Optical properties of spherical metallic QDs are well described by the Mie scattering theory.
The energy levels of a single particle in a quantum dot can be predicted using the particle in a box model in which the energies of states depend on the length of the box. For an exciton inside a quantum dot, there is also the Coulomb interaction between the negatively charged electron and the positively charged hole. By comparing the quantum dot's size to the exciton Bohr radius , three regimes can be defined. In the 'strong confinement regime', the quantum dot's radius is much smaller than the exciton Bohr radius, respectively the confinement energy dominates over the Coulomb interaction. [ 120 ] In the 'weak confinement' regime, the quantum dot is larger than the exciton Bohr radius, respectively the confinement energy is smaller than the Coulomb interactions between electron and hole. The regime where the exciton Bohr radius and confinement potential are comparable is called the 'intermediate confinement regime'. [ 121 ]
Therefore, the sum of these energies can be represented by Brus equation :
where μ is the reduced mass, a is the radius of the quantum dot, m e is the free electron mass, m h is the hole mass, and ε r is the size-dependent dielectric constant.
Although the above equations were derived using simplifying assumptions, they imply that the electronic transitions of the quantum dots will depend on their size. These quantum confinement effects are apparent only below the critical size. Larger particles do not exhibit this effect. This effect of quantum confinement on the quantum dots has been repeatedly verified experimentally [ 123 ] and is a key feature of many emerging electronic structures. [ 124 ]
The Coulomb interaction between confined carriers can also be studied by numerical means when results unconstrained by asymptotic approximations are pursued. [ 125 ]
Besides confinement in all three dimensions (that is, a quantum dot), other quantum confined semiconductors include:
A variety of theoretical frameworks exist to model optical, electronic, and structural properties of quantum dots. These may be broadly divided into quantum mechanical, semiclassical, and classical.
Quantum mechanical models and simulations of quantum dots often involve the interaction of electrons with a pseudopotential or random matrix . [ 126 ]
Semiclassical models of quantum dots frequently incorporate a chemical potential . For example, the thermodynamic chemical potential of an N -particle system is given by
whose energy terms may be obtained as solutions of the Schrödinger equation. The definition of capacitance,
with the potential difference
may be applied to a quantum dot with the addition or removal of individual electrons,
Then
is the quantum capacitance of a quantum dot, where we denoted by I ( N ) the ionization potential and by A ( N ) the electron affinity of the N -particle system. [ 127 ]
Classical models of electrostatic properties of electrons in quantum dots are similar in nature to the Thomson problem of optimally distributing electrons on a unit sphere.
The classical electrostatic treatment of electrons confined to spherical quantum dots is similar to their treatment in the Thomson, [ 128 ] or plum pudding model , of the atom. [ 129 ]
The classical treatment of both two-dimensional and three-dimensional quantum dots exhibit electron shell-filling behavior. A " periodic table of classical artificial atoms" has been described for two-dimensional quantum dots. [ 130 ] As well, several connections have been reported between the three-dimensional Thomson problem and electron shell-filling patterns found in naturally occurring atoms found throughout the periodic table. [ 131 ] This latter work originated in classical electrostatic modeling of electrons in a spherical quantum dot represented by an ideal dielectric sphere. [ 132 ]
For thousands of years, glassmakers were able to make colored glass by adding different dusts and powdered elements such as silver, gold and cadmium and then used different temperatures to produce shades of glass. In the 19th century, scientists started to understand how glass color depended on elements and heating-cooling techniques. It was also found that for the same element and preparation, the color depended on the dust particles' size. [ 133 ] [ 134 ]
Herbert Fröhlich in the 1930s first explored the idea that material properties can depend on the macroscopic dimensions of a small particle due to quantum size effects. [ 135 ]
The first quantum dots were synthesized in a glass matrix by Alexei A. Onushchenko and Alexey Ekimov in 1981 at the Vavilov State Optical Institute [ 136 ] [ 137 ] [ 138 ] [ 139 ] and independently in colloidal suspension [ 140 ] by Louis E. Brus team at Bell Labs in 1983. [ 141 ] [ 142 ] They were first theorized by Alexander Efros in 1982. [ 143 ] It was quickly identified that the optical changes that appeared for very small particles were due to quantum mechanical effects. [ 133 ]
The term quantum dot first appeared in a paper first authored by Mark Reed in 1986. [ 144 ] According to Brus, the term "quantum dot" was coined by Daniel S. Chemla [ de ] while they were working at Bell Labs. [ 145 ]
In 1993, David J. Norris, Christopher B. Murray and Moungi Bawendi at the Massachusetts Institute of Technology reported on a hot-injection synthesis method for producing reproducible quantum dots with well-defined size and with high optical quality. The method opened the door to the development of large-scale technological applications of quantum dots in a wide range of areas. [ 146 ] [ 133 ]
The Nobel Prize in Chemistry 2023 was awarded to Moungi Bawendi , Louis E. Brus and Alexey Ekimov "for the discovery and synthesis of quantum dots." [ 147 ] | https://en.wikipedia.org/wiki/Quantum_dot |
A quantum dot single-photon source is based on a single quantum dot placed in an optical cavity . It is an on-demand single-photon source. A laser pulse can excite a pair of carriers known as an exciton in the quantum dot. The decay of a single exciton due to spontaneous emission leads to the emission of a single photon. Due to interactions between excitons, the emission when the quantum dot contains a single exciton is energetically distinct from that when the quantum dot contains more than one exciton. Therefore, a single exciton can be deterministically created by a laser pulse and the quantum dot becomes a nonclassical light source that emits photons one by one and thus shows photon antibunching . The emission of single photons can be proven by measuring the second order intensity correlation function . The spontaneous emission rate of the emitted photons can be enhanced by integrating the quantum dot in an optical cavity . Additionally, the cavity leads to emission in a well-defined optical mode increasing the efficiency of the photon source.
With the growing interest in quantum information science since the beginning of the 21st century, research in different kinds of single-photon sources was growing. Early single-photon sources such as heralded photon sources [ 1 ] that were first reported in 1985 are based on non-deterministic processes. Quantum dot single-photon sources are on-demand. A single-photon source based on a quantum dot in a microdisk structure was reported in 2000. [ 2 ] Sources were subsequently embedded in different structures such as photonic crystals [ 3 ] or micropillars. [ 4 ] Adding distributed bragg reflectors (DBRs) allowed emission in a well-defined direction and increased emission efficiency. [ 5 ] Most quantum dot single-photon sources need to work at cryogenic temperatures , which is still a technical challenge. [ 5 ] The other challenge is to realize high-quality quantum dot single-photon sources at telecom wavelength for fiber telecommunication application. [ 6 ] The first report on Purcell-enhanced single-photon emission of a telecom-wavelength quantum dot in a two-dimensional photonic crystal cavity with a quality factor of 2,000 shows the enhancements of the emission rate and the intensity by five and six folds, respectively. [ 7 ]
Exciting an electron in a semiconductor from the valence band to the conduction band creates an excited state, a so-called exciton . The spontaneous radiative decay of this exciton results in the emission of a photon. Since a quantum dot has discrete energy levels, it can be achieved that there is never more than one exciton in the quantum dot simultaneously. Therefore, the quantum dot is an emitter of single photons. A key challenge in making a good single-photon source is to make sure that the emission from the quantum dot is collected efficiently. To do that, the quantum dot is placed in an optical cavity . The cavity can, for instance, consist of two DBRs in a micropillar (Fig. 1). The cavity enhances the spontaneous emission in a well-defined optical mode ( Purcell effect ), facilitating efficient guiding of the emission into an optical fiber. Furthermore, the reduced exciton lifetime Δ t {\displaystyle \Delta t} (see Fig. 2) reduces the significance of linewidth broadening due to noise.
The system can then be approximated by the Jaynes-Cummings model . In this model, the quantum dot only interacts with one single mode of the optical cavity. The frequency of the optical mode is well defined. This makes the photons indistinguishable if their polarization is aligned by a polarizer . The solution of the Jaynes-Cummings Hamiltonian is a vacuum Rabi oscillation . A vacuum Rabi oscillation of a photon interacting with an exciton is known as an exciton-polariton .
To eliminate the probability of the simultaneous emission of two photons it has to be made sure that there can only be one exciton in the cavity at one time. The discrete energy states in a quantum dot allow only one excitation. Additionally, the Rydberg blockade prevents the excitation of two excitons at the same space... [ 8 ] The electromagnetic interaction with the already existing exciton changes the energy for creating another exciton at the same space slightly. If the energy of the pump laser is tuned on resonance, the second exciton cannot be created.
Still, there is a small probability of having two excitations in the quantum dot at the same time. Two excitons confined in a small volume are called biexcitons . They interact with each other and thus slightly change their energy. Photons resulting from the decay of biexcitons have a different energy than photons resulting from the decay of excitons. They can be filtered out by letting the outgoing beam pass an optical filter . [ 9 ] The quantum dots can be excited both electrically and optically. [ 5 ] For optical pumping, a pulsed laser can be used for excitation of the quantum dots. In order to have the highest probability of creating an exciton, the pump laser is tuned on resonance. [ 10 ] This resembles a π {\displaystyle \pi } -pulse on the Bloch sphere . However, this way the emitted photons have the same frequency as the pump laser. A polarizer is needed to distinguish between them. [ 10 ] As the direction of polarization of the photons from the cavity is random, half of the emitted photons are blocked by this filter.
There are several ways to realize a quantum dot-cavity system that can act as a single-photon source. Typical cavity structures are micro-pillars, photonic crystal cavities, or tunable micro-cavities. Inside the cavity, different types of quantum dots can be used. The most widely used type are self-assembled InAs quantum dots grown in the Stranski-Krastanov growth mode, but other materials and growth methods such as local droplet etching [ 11 ] [ 12 ] have been used. A list of different experimental realizations is shown below:
Single photon sources exhibit antibunching . As photons are emitted one at a time, the probability of seeing two photons at the same time for an ideal source is 0. To verify the antibunching of a light source, one can measure the autocorrelation function g ( 2 ) ( τ ) {\displaystyle g^{(2)}(\tau )} . A photon source is antibunched if g ( 2 ) ( 0 ) {\displaystyle g^{(2)}(0)} ≤ g ( 2 ) ( τ ) {\displaystyle g^{(2)}(\tau )} . [ 25 ] For an ideal single photon source, g ( 2 ) ( 0 ) = 0 {\displaystyle g^{(2)}(0)=0} . Experimentally, g ( 2 ) ( τ ) {\displaystyle g^{(2)}(\tau )} is measured using the Hanbury Brown and Twiss effect . Using resonant excitation schemes, experimental values for g ( 2 ) ( 0 ) {\displaystyle g^{(2)}(0)} are typically in the regime of just a few percent. [ 10 ] [ 13 ] Values down to g ( 2 ) ( 0 ) = 7.5 × 10 − 5 {\displaystyle g^{(2)}(0)=7.5\times 10^{-5}} have been reached without resonant excitation. [ 26 ]
For applications the photons emitted by a single photon source must be indistinguishable . The theoretical solution of the Jaynes-Cummings Hamiltonian is a well-defined mode in which only the polarization is random. After aligning the polarization of the photons, their indistinguishability can be measured. For that, the Hong-Ou-Mandel effect is used. Two photons of the source are prepared so that they enter a 50:50 beam splitter at the same time from the two different input channels. A detector is placed on both exits of the beam splitter. Coincidences between the two detectors are measured. If the photons are indistinguishable, no coincidences should occur. [ 27 ] Experimentally, almost perfect indistinguishability is found. [ 13 ] [ 10 ]
Single-photon sources are of great importance in quantum communication science. They can be used for truly random number generators. [ 5 ] Single photons entering a beam splitter exhibit inherent quantum indeterminacy . Random numbers are used extensively in simulations using the Monte Carlo method .
Furthermore, single photon sources are essential in quantum cryptography . The BB84 [ 28 ] scheme is a provable secure quantum key distribution scheme. It works with a light source that perfectly emits only one photon at a time. Due to the no-cloning theorem , [ 29 ] no eavesdropping can happen without being noticed. The use of quantum randomness while writing the key prevents any patterns in the key that can be used to decipher the code.
Apart from that, single photon sources can be used to test some fundamental properties of quantum field theory . [ 1 ] | https://en.wikipedia.org/wiki/Quantum_dot_single-photon_source |
In condensed matter physics and quantum information theory , the quantum double model , proposed by Alexei Kitaev , is a lattice model that exhibits topological excitations. [ 1 ] This model can be regarded as a lattice gauge theory, and it has applications in many fields, like topological quantum computation , topological order , topological quantum memory , quantum error-correcting code , etc. The name "quantum double" come from the Drinfeld double of a finite groups and Hopf algebras . [ 2 ] The most well-known example is the toric code model , which is a special case of quantum double model by setting input group as cyclic group Z 2 {\displaystyle \mathbb {Z} _{2}} .
The input data for Kitaev quantum double is a finite group G {\displaystyle G} . Consider a directed lattice Σ {\displaystyle \Sigma } , we put a Hilbert space C [ G ] {\displaystyle \mathbb {C} [G]} spanned by group elements on each edge, there are four types of edge operators
L + g | h ⟩ = | g h ⟩ , L − g | h ⟩ = | h g − 1 ⟩ , {\displaystyle L_{+}^{g}|h\rangle =|gh\rangle ,L_{-}^{g}|h\rangle =|hg^{-1}\rangle ,}
T + g | h ⟩ = δ g , h | h ⟩ , T − g | h ⟩ = δ g − 1 , h | h ⟩ . {\displaystyle T_{+}^{g}|h\rangle =\delta _{g,h}|h\rangle ,T_{-}^{g}|h\rangle =\delta _{g^{-1},h}|h\rangle .}
For each vertex connecting to m {\displaystyle m} edges e 1 , … , e m {\displaystyle e_{1},\ldots ,e_{m}} , there is a vertex operator
A v = 1 | G | ∑ g ∈ G L g ( e 1 ) ⊗ … ⊗ L g ( e m ) . {\displaystyle A_{v}={\frac {1}{|G|}}\sum _{g\in G}L^{g}(e_{1})\otimes \ldots \otimes L^{g}(e_{m}).}
Notice each edge has an orientation: when v {\displaystyle v} is the starting point of e k {\displaystyle e_{k}} , the operator is set as L − {\displaystyle L_{-}} , otherwise, it is set as L + {\displaystyle L_{+}} .
For each face surrounded by m {\displaystyle m} edges e 1 , … , e m {\displaystyle e_{1},\ldots ,e_{m}} , there is a face operator
B f = ∑ h 1 ⋯ h m = 1 G ∏ k = 1 m T h k ( e k ) . {\displaystyle B_{f}=\sum _{h_{1}\cdots h_{m}=1_{G}}\prod _{k=1}^{m}T^{h_{k}}(e_{k}).}
Similar to the vertex operator, due to the orientation of the edge, when face f {\displaystyle f} is on the right-hand side when traversing the positive direction of e {\displaystyle e} , we set T + {\displaystyle T_{+}} ; otherwise, we set T − {\displaystyle T_{-}} in the above expression. Also, note that the order of edges surrounding the face is assumed to be counterclockwise.
The lattice Hamiltonian of quantum double model is given by
H = − ∑ v A v − ∑ f B f . {\displaystyle H=-\sum _{v}A_{v}-\sum _{f}B_{f}.}
Both of A v {\displaystyle A_{v}} and B f {\displaystyle B_{f}} are Hermitian projectors, they are stabilizer when regard the model is a quantum error correcting code.
The topological excitations of the model is characterized by the representations of the quantum double of finite group G {\displaystyle G} . The anyon types are given by irreducible representations. For the lattice model, the topological excitations are created by ribbon operators. [ 1 ] [ 3 ]
The gapped boundary theory of quantum double model can be constructed based on subgroups of G {\displaystyle G} . [ 4 ] [ 5 ] [ 6 ] There is a boundary-bulk duality for this model.
The topological excitation of the model is equivalent to that of the Levin-Wen string-net model with input given by the representation category of finite group G {\displaystyle G} .
The quantum double model can be generalized to the case where the input data is given by a C* Hopf algebra . [ 7 ] In this case, the face and vertex operators are constructed using the comultiplication of Hopf algebra. For each vertex, the Haar integral of the input Hopf algebra is used to construct the vertex operator. For each face, the Haar integral of the dual Hopf algebra of the input Hopf algebra is used to construct the face operator.
The topological excitation are created by ribbon operators. [ 8 ] [ 9 ] [ 5 ]
A more general case arises when the input data is chosen as a weak Hopf algebra, resulting in the weak Hopf quantum double model. [ 10 ] [ 11 ] | https://en.wikipedia.org/wiki/Quantum_double_model |
In physics, quantum dynamics is the quantum version of classical dynamics . Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of quantum mechanics . [ 1 ] [ 2 ] Quantum dynamics is relevant for burgeoning fields, such as quantum computing and atomic optics .
In mathematics, quantum dynamics is the study of the mathematics behind quantum mechanics . [ 3 ] Specifically, as a study of dynamics , this field investigates how quantum mechanical observables change over time. Most fundamentally, this involves the study of one-parameter automorphisms of the algebra of all bounded operators on the Hilbert space of observables (which are self-adjoint operators). These dynamics were understood as early as the 1930s, after Wigner , Stone , Hahn and Hellinger worked in the field. Recently, [ when? ] mathematicians in the field have studied irreversible quantum mechanical systems on von Neumann algebras . [ 4 ]
Equations to describe quantum systems can be seen as equivalent to that of classical dynamics on a macroscopic scale , except for the important detail that the variables don't follow the commutative laws of multiplication. [ 5 ] Hence, as a fundamental principle, these variables are instead described as " q-numbers ", conventionally represented by operators or Hermitian matrices on a Hilbert space . [ 6 ] Indeed, the state of the system in the atomic and subatomic scale is described not by dynamic variables with specific numerical values, but by state functions that are dependent on the c-number time. In this realm of quantum systems, the equation of motion governing dynamics heavily relies on the Hamiltonian , also known as the total energy. Therefore, to anticipate the time evolution of the system, one only needs to determine the initial condition of the state function |Ψ(t) and its first derivative with respect to time. [ 7 ]
For example, quasi-free states and automorphisms are the Fermionic counterparts of classical Gaussian measures [ 8 ] ( Fermions' descriptors are Grassmann operators). [ 6 ] | https://en.wikipedia.org/wiki/Quantum_dynamics |
The term quantum efficiency ( QE ) may apply to incident photon to converted electron ( IPCE ) ratio [ 1 ] of a photosensitive device , or it may refer to the TMR effect of a magnetic tunnel junction.
This article deals with the term as a measurement of a device's electrical sensitivity to light. In a charge-coupled device (CCD) or other photodetector, it is the ratio between the number of charge carriers collected at either terminal and the number of photons hitting the device's photoreactive surface. As a ratio, QE is dimensionless, but it is closely related to the responsivity , which is expressed in amps per watt . Since the energy of a photon is inversely proportional to its wavelength , QE is often measured over a range of different wavelengths to characterize a device's efficiency at each photon energy level. For typical semiconductor photodetectors, QE drops to zero for photons whose energy is below the band gap . A photographic film typically has a QE of much less than 10%, [ 2 ] while CCDs can have a QE of well over 90% at some wavelengths.
A solar cell 's quantum efficiency value indicates the amount of current that the cell will produce when irradiated by photons of a particular wavelength. If the cell's quantum efficiency is integrated over the whole solar electromagnetic spectrum , one can evaluate the amount of current that the cell will produce when exposed to sunlight. The ratio between this energy-production value and the highest possible energy-production value for the cell (i.e., if the QE were 100% over the whole spectrum) gives the cell's overall energy conversion efficiency value. Note that in the event of multiple exciton generation (MEG), quantum efficiencies of greater than 100% may be achieved since the incident photons have more than twice the band gap energy and can create two or more electron-hole pairs per incident photon.
Two types of quantum efficiency of a solar cell are often considered:
The IQE is always larger than the EQE in the visible spectrum. A low IQE indicates that the active layer of the solar cell is unable to make good use of the photons, most likely due to poor carrier collection efficiency. To measure the IQE, one first measures the EQE of the solar device, then measures its transmission and reflection, and combines these data to infer the IQE. EQE = electrons/sec photons/sec = (current) / (charge of one electron) ( total power of photons ) / ( energy of one photon ) {\displaystyle {\text{EQE}}={\frac {\text{electrons/sec}}{\text{photons/sec}}}={\frac {{\text{(current)}}/{\text{(charge of one electron)}}}{({\text{total power of photons}})/({\text{energy of one photon}})}}} IQE = electrons/sec absorbed photons/sec = EQE 1-Reflection-Transmission {\displaystyle {\text{IQE}}={\frac {\text{electrons/sec}}{\text{absorbed photons/sec}}}={\frac {\text{EQE}}{\text{1-Reflection-Transmission}}}}
The external quantum efficiency therefore depends on both the absorption of light and the collection of charges. Once a photon has been absorbed and has generated an electron-hole pair, these charges must be separated and collected at the junction. A "good" material avoids charge recombination. Charge recombination causes a drop in the external quantum efficiency.
The ideal quantum efficiency graph has a square shape , where the QE value is fairly constant across the entire spectrum of wavelengths measured. However, the QE for most solar cells is reduced because of the effects of recombination, where charge carriers are not able to move into an external circuit. The same mechanisms that affect the collection probability also affect the QE. For example, modifying the front surface can affect carriers generated near the surface. Highly doped front surface layers can also cause 'free carrier absorption' which reduces QE in the longer wavelengths. [ 3 ] And because high-energy (blue) light is absorbed very close to the surface, considerable recombination at the front surface will affect the "blue" portion of the QE. Similarly, lower energy (green) light is absorbed in the bulk of a solar cell, and a low diffusion length will affect the collection probability from the solar cell bulk, reducing the QE in the green portion of the spectrum. Generally, solar cells on the market today do not produce much electricity from ultraviolet and infrared light (<400 nm and >1100 nm wavelengths, respectively); these wavelengths of light are either filtered out or are absorbed by the cell, thus heating the cell. That heat is wasted energy, and could damage the cell. [ 4 ]
Quantum efficiency (QE) is the fraction of photon flux that contributes to the photocurrent in a photodetector or
a pixel. Quantum efficiency is one of the most important parameters used to evaluate the quality of a detector and is often called the spectral response to reflect its wavelength dependence. It is defined as the number of signal electrons created per incident photon. In some cases it can exceed 100% (i.e. when more than one electron is created per incident photon).
Conventional measurement of the EQE will give the efficiency of the overall device. However it is often useful to have a map of the EQE over large area of the device. This mapping provides an efficient way to visualize the homogeneity and/or the defects in the sample. It was realized by researchers from the Institute of Researcher and Development on Photovoltaic Energy (IRDEP) who calculated the EQE mapping from electroluminescence measurements taken with a hyperspectral imager. [ 5 ] [ 6 ]
Spectral responsivity is a similar measurement, but it has different units: amperes per watt (A/W); (i.e. how much current comes out of the device per unit of incident light power ). [ 7 ] Responsivity is ordinarily specified for monochromatic light (i.e. light of a single wavelength). [ citation needed ] Both the quantum efficiency and the responsivity are functions of the photons' wavelength (indicated by the subscript λ).
To convert from responsivity ( R λ , in A/W) to QE λ [ 8 ] (on a scale 0 to 1): Q E λ = R λ λ × h c e ≈ R λ λ × ( 1240 W ⋅ n m / A ) {\displaystyle QE_{\lambda }={\frac {R_{\lambda }}{\lambda }}\times {\frac {hc}{e}}\approx {\frac {R_{\lambda }}{\lambda }}{\times }(1240\;\mathrm {W\cdot {nm}/A} )} where λ is the wavelength in nm , h is the Planck constant , c is the speed of light in vacuum, and e is the elementary charge . Note that the unit W/A (watts per ampere) is equivalent to V (volts).
Q E λ = η = N e N ν {\displaystyle QE_{\lambda }=\eta ={\frac {N_{e}}{N_{\nu }}}} where N e {\displaystyle N_{e}} = number of electrons produced, N ν {\displaystyle N_{\nu }} = number of photons absorbed. N ν t = Φ o λ h c {\displaystyle {\frac {N_{\nu }}{t}}=\Phi _{o}{\frac {\lambda }{hc}}}
Assuming each photon absorbed in the depletion layer produces a viable electron-hole pair, and all other photons do not, N e t = Φ ξ λ h c {\displaystyle {\frac {N_{e}}{t}}=\Phi _{\xi }{\frac {\lambda }{hc}}} where t is the measurement time (in seconds), Φ o {\displaystyle \Phi _{o}} = incident optical power in watts, Φ ξ {\displaystyle \Phi _{\xi }} = optical power absorbed in depletion layer, also in watts. | https://en.wikipedia.org/wiki/Quantum_efficiency |
The scientific school of Quantum electrochemistry began to form in the 1960s under Revaz Dogonadze . Generally speaking, the field comprises the notions arising in electrodynamics , quantum mechanics , and electrochemistry ; and so is studied by a very large array of different professional researchers. The fields they reside in include, chemical , electrical and mechanical engineering , chemistry and physics .
More specifically, quantum electrochemistry is the application of quantum mechanical tools such as density functional theory to the study of electrochemical processes, including electron transfer at electrodes. [ 1 ] It also includes Marcus theory and quantum rate theory, [ 2 ] the latter being a method of describing electrochemistry using first principle quantum mechanics and concepts of conductance quantum and quantum capacitance .
The first development of "quantum electrochemistry" is somewhat difficult to pin down. This is not very surprising, since the development of quantum mechanics to chemistry can be summarized as the application of quantum wave theory models to atoms and molecules . This being the case, electrochemistry , which is particularly concerned with the electronic states of some particular system, is already, by its nature, tied into the quantum mechanical model of the electron in quantum chemistry. There were proponents of quantum electrochemistry, who applied quantum mechanics to electrochemistry with unusual zeal, clarity, and precision. Among them were Revaz Dogonadze and his co-workers. They developed one of the early quantum mechanical models for proton transfer reactions in chemical systems. Dogonadze is a particularly celebrated promoter of quantum electrochemistry and is also credited with forming an international summer school of quantum electrochemistry centered in Yugoslavia . He was the main author of the Quantum-Mechanical Theory of Kinetics of the Elementary Act of Chemical, Electrochemical and Biochemical Processes in Polar Liquids . Another important contributor is Rudolph A. Marcus , who won the Nobel Prize in Chemistry in 1992 for his Theory of Electron Transfer Reactions in Chemical Systems . Recently, Marcus theory has been shown to be part of a more general concept associated with the quantum rate theory, a theory that predicts the rate of electron transfer (electrochemistry being a particular case) based on the uses of conductance quantum and quantum capacitance concepts. | https://en.wikipedia.org/wiki/Quantum_electrochemistry |
Quantum engineering is the development of technology that capitalizes on the laws of quantum mechanics. This type of engineering uses quantum mechanics to develop technologies such as quantum sensors and quantum computers .
Devices that rely on quantum mechanical effects such as lasers , MRI imagers and transistors have revolutionized many areas of technology. New technologies are being developed that rely on phenomena such as quantum coherence and on progress achieved in the last century in understanding and controlling atomic-scale systems. Quantum mechanical effects are used as a resource in novel technologies with far-reaching applications, including quantum sensors [ 1 ] [ 2 ] and novel imaging techniques, [ 3 ] secure communication ( quantum internet ) [ 4 ] [ 5 ] [ 6 ] and quantum computing. [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ]
The field of quantum technology was explored in a 1997 book by Gerard J. Milburn . [ 12 ] It was then followed by a 2003 article by Milburn and Jonathan P. Dowling , [ 13 ] and a separate publication by David Deutsch on the same year. [ 14 ]
The application of quantum mechanics was evident in several technologies. These include laser systems, transistors and semiconductor devices, as well as other devices such as MRI imagers. The UK Defence Science and Technology Laboratory (DSTL) grouped these devices as 'quantum 1.0' to differentiate them from what it dubbed as 'quantum 2.0'. This is a definition of the class of devices that actively create, manipulate, and read out quantum states of matter using the effects of superposition and entanglement. [ 15 ]
From 2010 onwards, multiple governments have established programmes to explore quantum technologies, [ 16 ] such as the UK National Quantum Technologies Programme, [ 17 ] which created four quantum 'hubs'. These hubs are found at the Centre for Quantum Technologies in Singapore, and QuTech, a Dutch center to develop a topological quantum computer. [ 18 ] In 2016, the European Union introduced the Quantum Technology Flagship , [ 19 ] [ 20 ] a €1 Billion, 10-year-long megaproject , similar in size to earlier European Future and Emerging Technologies Flagship projects. [ 21 ] [ 22 ] In December 2018, the United States passed the National Quantum Initiative Act , which provides a US$1 billion annual budget for quantum research. [ 23 ] China is building the world's largest quantum research facility with a planned investment of 76 billion Yuan (approx. €10 Billion). [ 24 ] [ 25 ] Indian government has also invested 8000 crore Rupees (approx. US$1.02 Billion) over 5-years to boost quantum technologies under its National Quantum Mission. [ 26 ]
In the private sector, large companies have made multiple investments in quantum technologies. Organizations such as Google , D-wave systems , and University of California Santa Barbara [ 27 ] have formed partnerships and investments to develop quantum technology.
Quantum secure communication is a method that is expected to be 'quantum safe' in the advent of quantum computing systems that could break current cryptography systems using methods such as Shor's algorithm . These methods include quantum key distribution (QKD) , a method of transmitting information using entangled light in a way that makes any interception of the transmission obvious to the user. Another method is the quantum random number generator, which is capable of producing truly random numbers unlike non-quantum algorithms that merely imitate randomness. [ 28 ]
Quantum computers are expected to have a number of important uses in computing fields such as optimization and machine learning. They are perhaps best known for their expected ability to carry out Shor's algorithm, which can be used to factorize large numbers and is an important process in the securing of data transmissions.
Quantum simulators are types of quantum computers intended to simulate a real world system, such as a chemical compound. [ 29 ] [ 30 ] Quantum simulators are simpler to build as opposed to general purpose quantum computers because complete control over every component is not necessary. [ 29 ] Current quantum simulators under development include ultracold atoms in optical lattices, trapped ions, arrays of superconducting qubits, and others. [ 29 ]
Quantum sensors are expected to have a number of applications in a wide variety of fields including positioning systems, communication technology, electric and magnetic field sensors, gravimetry [ 31 ] as well as geophysical areas of research such as civil engineering [ 32 ] and seismology.
Quantum engineering is evolving into its own engineering discipline. The quantum industry requires a quantum-literate workforce, a missing resource at the moment. Currently, scientists in the field of quantum technology have mostly either a physics or engineering background and have acquired their ”quantum engineering skills” by experience. A survey of more than twenty companies aimed to understand the scientific, technical, and “soft” skills required of new hires into the quantum industry. Results show that companies often look for people that are familiar with quantum technologies and simultaneously possess excellent hands-on lab skills. [ 33 ]
Several technical universities have launched education programs in this domain. For example, ETH Zurich has initiated a Master of Science in Quantum Engineering, a joint venture between the electrical engineering department (D-ITET) and the physics department (D-PHYS), EPFL offers a dedicated Master's program in Quantum Science and Engineering, combining coursework in quantum physics and engineering with research opportunities, and the University of Waterloo has launched integrated postgraduate engineering programs within the Institute for Quantum Computing . [ 34 ] [ 35 ] Similar programs are being pursued at Delft University , Technical University of Munich , MIT , CentraleSupélec and other technical universities.
In the realm of undergraduate studies, opportunities for specialization are sparse. Nevertheless, some institutions have begun to offer programs. The Université de Sherbrooke offers a Bachelor of Science in quantum information, [ 36 ] University of Waterloo offers a quantum specialization in its electrical engineering program, and the University of New South Wales offers a bachelor of quantum engineering. [ 37 ] A report on the development of this bachelor degree has been published in IEEE Transactions on Quantum Engineering. [ 38 ]
Students are trained in signal and information processing, optoelectronics and photonics, integrated circuits (bipolar, CMOS ) and electronic hardware architectures ( VLSI , FPGA , ASIC ). In addition, they are exposed to emerging applications such as quantum sensing, quantum communication and cryptography and quantum information processing. They learn the principles of quantum simulation and quantum computing, and become familiar with different quantum processing platforms, such as trapped ions , and superconducting circuits . Hands-on laboratory projects help students to develop the technical skills needed for the practical realization of quantum devices, consolidating their education in quantum science and technologies. | https://en.wikipedia.org/wiki/Quantum_engineering |
In quantum chaos , a branch of mathematical physics , quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to initial conditions. Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to energy eigenstates of a quantized ergodic Hamiltonian tend to a uniform distribution in the classical phase space . This is consistent with the intuition that the flows of ergodic systems are equidistributed in phase space. By contrast, classical completely integrable systems generally have periodic orbits in phase space, and this is exhibited in a variety of ways in the high-energy limit of the eigenstates: typically, some form of concentration occurs in the semiclassical limit ℏ → 0 {\displaystyle \hbar \rightarrow 0} .
The model case of a Hamiltonian is the geodesic Hamiltonian on the cotangent bundle of a compact Riemannian manifold . The quantization of the geodesic flow is given by the fundamental solution of the Schrödinger equation
where Δ {\displaystyle {\sqrt {\Delta }}} is the square root of the Laplace–Beltrami operator . The quantum ergodicity theorem of Shnirelman 1974, Zelditch , and Yves Colin de Verdière states that a compact Riemannian manifold whose unit tangent bundle is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the n th eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as n → ∞ in a subset of the natural numbers of natural density equal to one. Quantum ergodicity can be formulated as a non-commutative analogue of the classical ergodicity ( T. Sunada ).
Since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space. Thus, when translating the concept of ergodicity to the quantum realm, it is natural to assume that the eigenstates of the quantum chaotic system would fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit ℏ → 0 {\displaystyle \hbar \rightarrow 0} . The quantum ergodicity theorems of Shnirelman, Zelditch , and Yves Colin de Verdière proves that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. However, the quantum ergodicity theorem leaves open the possibility of eigenfunctions become sparse with serious holes as ℏ → 0 {\displaystyle \hbar \rightarrow 0} , leaving large but not macroscopic gaps on the energy manifolds in the phase space. In particular, the theorem allows the existence of a subset of macroscopically nonergodic states which on the other hand must approach zero measure, i.e., the contribution of this set goes towards zero percent of all eigenstates when ℏ → 0 {\displaystyle \hbar \rightarrow 0} . [ 5 ]
For example, the theorem do not exclude quantum scarring, as the phase space volume of the scars also gradually vanishes in this limit. [ 1 ] [ 5 ] [ 6 ] [ 2 ] A quantum eigenstate is scarred by periodic orbit if its probability density is on the classical invariant manifolds near and all along that periodic orbit is systematically enhanced above the classical, statistically expected density along that orbit. [ 5 ] In a simplified manner, a quantum scar refers to an eigenstate of whose probability density is enhanced in the neighborhood of a classical periodic orbit when the corresponding classical system is chaotic. In conventional scarring, the responsive periodic orbit is unstable. [ 1 ] [ 5 ] [ 6 ] [ 2 ] The instability is a decisive point that separates quantum scars from a more trivial finding that the probability density is enhanced near stable periodic orbits due to the Bohr's correspondence principle . The latter can be viewed as a purely classical phenomenon, whereas in the former quantum interference is important. On the other hand, in the perturbation-induced quantum scarring, [ 3 ] [ 7 ] [ 8 ] [ 9 ] [ 4 ] some of the high-energy eigenstates of a locally perturbed quantum dot contain scars of short periodic orbits of the corresponding unperturbed system. Even though similar in appearance to ordinary quantum scars, these scars have a fundamentally different origin., [ 3 ] [ 7 ] [ 4 ] In this type of scarring, there are no periodic orbits in the perturbed classical counterpart or they are too unstable to cause a scar in a conventional sense. Conventional and perturbation-induced scars are both a striking visual example of classical-quantum correspondence and of a quantum suppression of chaos (see the figure). In particular, scars are a significant correction to the assumption that the corresponding eigenstates of a classically chaotic Hamiltonian are only featureless and random. In some sense, scars can be considered as an eigenstate counterpart to the quantum ergodicity theorem of how short periodic orbits provide corrections to the universal random matrix theory eigenvalue statistics. | https://en.wikipedia.org/wiki/Quantum_ergodicity |
Quantum evolution is a component of George Gaylord Simpson 's multi-tempoed theory of evolution proposed to explain the rapid emergence of higher taxonomic groups in the fossil record . According to Simpson, evolutionary rates differ from group to group and even among closely related lineages. These different rates of evolutionary change were designated by Simpson as bradytelic (slow tempo), horotelic (medium tempo), and tachytelic (rapid tempo).
Quantum evolution differed from these styles of change in that it involved a drastic shift in the adaptive zones of certain classes of animals. The word " quantum " therefore refers to an "all-or-none reaction", where transitional forms are particularly unstable, and thereby perish rapidly and completely. [ 1 ] Although quantum evolution may happen at any taxonomic level, [ 2 ] it plays a much larger role in "the origin taxonomic units of relatively high rank, such as families , orders , and classes ." [ 3 ]
Usage of the phrase "quantum evolution" in plants was apparently first articulated by Verne Grant in 1963 (pp. 458-459). [ 4 ] He cited an earlier 1958 paper by Harlan Lewis and Peter H. Raven , [ 5 ] wherein Grant asserted that Lewis and Raven gave a "parallel" definition of quantum evolution as defined by Simpson. Lewis and Raven postulated that species in the Genus Clarkia had a mode of speciation that resulted
...as a consequence of a rapid reorganization of the chromosomes due to the presence, at some time, of a genotype conducive to extensive chromosome breakage. A similar mode of origin by rapid reorganization of the chromosomes is suggested for the derivation of other species of Clarkia . In all of these examples the derivative populations grow adjacent to the parental species, which they resemble closely in morphology, but from which they are reproductively isolated because of multiple structural differences in their chromosomes. The spatial relationship of each parental species and its derivative suggests that differentiation has been recent. The repeated occurrence of the same pattern of differentiation in Clarkia suggests that a rapid reorganization of chromosomes has been an important mode of evolution in the genus. This rapid reorganization of the chromosomes is comparable to the systemic mutations proposed by Goldschmidt as a mechanism of macroevolution . In Clarkia , we have not observed marked changes in physiology and pattern of development that could be described as macroevolution. Reorganization of the genomes may, however, set the stage for subsequent evolution along a very different course from that of the ancestral populations [ 5 ]
Harlan Lewis refined this concept in a 1962 paper [ 6 ] where he coined the term "Catastrophic Speciation" to describe this mode of speciation, since he theorized that the reductions in population size and consequent inbreeding that led to chromosomal rearrangements occurred in small populations that were subject to severe drought.
Leslie D. Gottlieb in his 2003 summary of the subject in plants stated [ 7 ]
we can define quantum speciation as the budding off of a new and very different daughter species from a semi-isolated peripheral population of the ancestral species in a cross-fertilizing organism...as compared with geographical speciation, which is a gradual and conservative process, quantum speciation is rapid and radical in its phenotypic or genotypic effects or both.
Gottlieb did not believe that sympatric speciation required disruptive selection to form a reproductive isolating barrier, as defined by Grant, and in fact Gottlieb stated that requiring disruptive selection was "unnecessarily restrictive" [ 8 ] in identifying cases of sympatric speciation. In this 2003 paper Gottlieb summarized instances of quantum evolution in the plant species Clarkia , Layia , and Stephanomeria .
According to Simpson (1944), quantum evolution resulted from Sewall Wright 's model of random genetic drift . Simpson believed that major evolutionary transitions would arise when small populations, that were isolated and limited from gene flow , would fixate upon unusual gene combinations. This "inadaptive phase" (caused by genetic drift) would then (by natural selection) drive a deme population from one stable adaptive peak to another on the adaptive fitness landscape . However, in his Major Features of Evolution (1953) Simpson wrote that this mechanism was still controversial:
"whether prospective adaptation as prelude to quantum evolution arises adaptively or inadaptively. It was concluded above that it usually arises adaptively . . . . The precise role of, say, genetic drift in this process thus is largely speculative at present. It may have an essential part or none. It surely is not involved in all cases of quantum evolution, but there is a strong possibility that it is often involved. If or when it is involved, it is an initiating mechanism. Drift can only rarely, and only for lower categories, have completed the transition to a new adaptive zone." [ 9 ]
This preference for adaptive over inadaptive forces led Stephen Jay Gould to call attention to the "hardening of the Modern Synthesis", a trend in the 1950s where adaptationism took precedence over the pluralism of mechanisms common in the 1930s and 40s. [ 10 ]
Simpson considered quantum evolution his crowning achievement, being "perhaps the most important outcome of [my] investigation, but also the most controversial and hypothetical." [ 3 ] | https://en.wikipedia.org/wiki/Quantum_evolution |
Quantum feedback or quantum feedback control is a class of methods to prepare and manipulate a quantum system in which that system's quantum state or trajectory is used to evolve the system towards some desired outcome. Just as in the classical case, feedback occurs when outputs from the system are used as inputs that control the dynamics (e.g. by controlling the Hamiltonian of the system). The feedback signal is typically filtered or processed in a classical way, which is often described as measurement based feedback . However, quantum feedback also allows the possibility of maintaining the quantum coherence of the output as the signal is processed (via unitary evolution ), which has no classical analogue. [ 1 ] [ 2 ] [ 3 ]
In the closed loop quantum control, the feedback may be entirely dynamical (that is, the plant and controller form a single dynamical system and the controller with the two influencing each other through direct interaction). This is named Coherent Control. Alternatively, the feedback may be entirely information theoretic insofar as the controller gains information about the plant due to measurement of the plant. This is measurement-based control.
Unlike measurement based feedback, where the quantum state is measured (causing it to collapse) and control is conditioned on the classical measurement outcome, coherent feedback maintains the full quantum state and implements deterministic, non-destructive operations on the state, using fully quantum devices.
One example is a mirror, reflecting photons (the quantum states) back to the emitter. [ 4 ] | https://en.wikipedia.org/wiki/Quantum_feedback |
In theoretical physics , quantum field theory ( QFT ) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics . [ 1 ] : xi QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles . The current standard model of particle physics is based on QFT.
Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons , culminating in the first quantum field theory— quantum electrodynamics . A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inability to describe the weak and strong interactions , to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory.
Quantum field theory results from the combination of classical field theory , quantum mechanics , and special relativity . [ 1 ] : xi A brief overview of these theoretical precursors follows.
The earliest successful classical field theory is one that emerged from Newton's law of universal gravitation , despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica . The force of gravity as described by Isaac Newton is an " action at a distance "—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley , however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact". [ 2 ] : 4 It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a vector in the case of gravitational field ) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick. [ 3 ] : 18
Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day. [ 2 ] [ 4 ] : 301 [ 5 ] : 2
The theory of classical electromagnetism was completed in 1864 with Maxwell's equations , which described the relationship between the electric field , the magnetic field , electric current , and electric charge . Maxwell's equations implied the existence of electromagnetic waves , a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light . Action-at-a-distance was thus conclusively refuted. [ 2 ] : 19
Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra , nor for the distribution of blackbody radiation in different wavelengths. [ 6 ] Max Planck 's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit electromagnetic radiation , as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators . This process of restricting energies to discrete values is called quantization. [ 7 ] : Ch.2 Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect , that light is composed of individual packets of energy called photons (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles. [ 6 ]
In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave–particle duality , that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. [ 6 ] Uniting these scattered ideas, a coherent discipline, quantum mechanics , was formulated between 1925 and 1926, with important contributions from Max Planck , Louis de Broglie , Werner Heisenberg , Max Born , Erwin Schrödinger , Paul Dirac , and Wolfgang Pauli . [ 3 ] : 22–23
In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity , built on Maxwell's electromagnetism. New rules, called Lorentz transformations , were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred. [ 3 ] : 19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying quantum mechanics could explain the stimulated emission of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain spontaneous emission , where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to linear operators . [ 6 ]
Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s. [ 8 ] : 1
Through the works of Born, Heisenberg, and Pascual Jordan in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via canonical quantization by treating the electromagnetic field as a set of quantum harmonic oscillators . [ 8 ] : 1 With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world. [ 3 ] : 22
In his seminal 1927 paper The quantum theory of the emission and absorption of radiation , Dirac coined the term quantum electrodynamics (QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric current density and the electromagnetic vector potential . Using first-order perturbation theory , he successfully explained the phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the ground state ). Therefore, even in a perfect vacuum , there remains an oscillating electromagnetic field having zero-point energy . It is this quantum fluctuation of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the scattering of photons, resonance fluorescence and non-relativistic Compton scattering . Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations. [ 6 ] : 71
In 1928, Dirac wrote down a wave equation that described relativistic electrons: the Dirac equation . It had the following important consequences: the spin of an electron is 1/2; the electron g -factor is 2; it led to the correct Sommerfeld formula for the fine structure of the hydrogen atom ; and it could be used to derive the Klein–Nishina formula for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation. [ 6 ] : 71–72
The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and quantum fields (such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the excited states of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, Eugene Wigner , Heisenberg, Pauli, and Enrico Fermi discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for beta decay known as Fermi's interaction . Atomic nuclei do not contain electrons per se , but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom. [ 3 ] : 22–23
It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of antimatter . Indeed, the evidence for positrons was discovered in 1932 by Carl David Anderson in cosmic rays . With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called pair production ; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the Dirac hole theory . [ 6 ] : 72 [ 3 ] : 23 QFT naturally incorporated antiparticles in its formalism. [ 3 ] : 24
Robert Oppenheimer showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron self-energy and the vacuum zero-point energy of the electron and photon fields, [ 6 ] suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta. [ 3 ] : 25 It was not until 20 years later that a systematic approach to remove such infinities was developed.
A series of papers was published between 1934 and 1938 by Ernst Stueckelberg that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community. [ 6 ]
Faced with these infinities, John Archibald Wheeler and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called S-matrix theory . Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of observables ( e.g. the energy of an atom) in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945, Richard Feynman and Wheeler daringly suggested abandoning QFT altogether and proposed action-at-a-distance as the mechanism of particle interactions. [ 3 ] : 26
In 1947, Willis Lamb and Robert Retherford measured the minute difference in the 2 S 1/2 and 2 P 1/2 energy levels of the hydrogen atom, also called the Lamb shift . By ignoring the contribution of photons whose energy exceeds the electron mass, Hans Bethe successfully estimated the numerical value of the Lamb shift. [ 6 ] [ 3 ] : 28 Subsequently, Norman Myles Kroll , Lamb, James Bruce French , and Victor Weisskopf again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations. [ 6 ]
The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by Julian Schwinger , Richard Feynman , Freeman Dyson , and Shinichiro Tomonaga . The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as renormalization and can be applied to arbitrary order in perturbation theory. [ 6 ] As Tomonaga said in his Nobel lecture:
Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with [the] Americans'. [ 9 ]
By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment (the deviation of the electron g -factor from 2) and vacuum polarization . These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities". [ 6 ]
At the same time, Feynman introduced the path integral formulation of quantum mechanics and Feynman diagrams . [ 8 ] : 2 The latter can be used to visually and intuitively organize and to help compute terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the scattering amplitude of the interaction represented by the diagram. [ 1 ] : 5
It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework. [ 8 ] : 2
Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades. [ 3 ] : 30
The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the Fermi theory of the weak interaction , are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities. [ 3 ] : 30
The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the coupling constant , in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the fine-structure constant α ≈ 1/137 , which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the strong interaction is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods. [ 3 ] : 31
With these difficulties looming, many theorists began to turn away from QFT. Some focused on symmetry principles and conservation laws , while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations. [ 3 ] : 31
Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory, [ 10 ] : 454 but in 1951 [ 11 ] [ 12 ] he found a way around the problem of the infinities with a new method using external sources as currents coupled to gauge fields. [ 13 ] Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the classical process of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his source theory in 1966 [ 14 ] then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields. [ 15 ] [ 16 ] [ 17 ] Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed. [ 15 ]
In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general. [ 18 ] Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities. [ 10 ] : 467
Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury. [ 19 ] The neglect of source theory by the physics community was a major disappointment for Schwinger:
The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger [ 15 ]
See " the shoes incident " between J. Schwinger and S. Weinberg . [ 10 ]
In 1954, Yang Chen-Ning and Robert Mills generalized the local symmetry of QED, leading to non-Abelian gauge theories (also known as Yang–Mills theories), which are based on more complicated local symmetry groups . [ 20 ] : 5 In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of " charge " interact via the exchange of massless gauge bosons . Unlike photons, these gauge bosons themselves carry charge. [ 3 ] : 32 [ 21 ]
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, Abdus Salam and John Clive Ward arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable. [ 22 ]
Peter Higgs , Robert Brout , François Englert , Gerald Guralnik , Carl Hagen , and Tom Kibble proposed in their famous Physical Review Letters papers that the gauge symmetry in Yang–Mills theories could be broken by a mechanism called spontaneous symmetry breaking , through which originally massless gauge bosons could acquire mass. [ 20 ] : 5–6
By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, Steven Weinberg wrote down in 1967 a theory describing electroweak interactions between all leptons and the effects of the Higgs boson . His theory was at first mostly ignored, [ 22 ] [ 20 ] : 6 until it was brought back to light in 1971 by Gerard 't Hooft 's proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to quarks in 1970 by Glashow, John Iliopoulos , and Luciano Maiani , marking its completion. [ 22 ]
Harald Fritzsch , Murray Gell-Mann , and Heinrich Leutwyler discovered in 1971 that certain phenomena involving the strong interaction could also be explained by non-Abelian gauge theory. Quantum chromodynamics (QCD) was born. In 1973, David Gross , Frank Wilczek , and Hugh David Politzer showed that non-Abelian gauge theories are " asymptotically free ", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) [ 20 ] : 11 Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible. [ 3 ] : 32
These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the Standard Model of elementary particles. [ 23 ] The Standard Model successfully describes all fundamental interactions except gravity , and its many predictions have been met with remarkable experimental confirmation in subsequent decades. [ 8 ] : 3 The Higgs boson , central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at CERN , marking the complete verification of the existence of all constituents of the Standard Model. [ 24 ]
The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The 't Hooft–Polyakov monopole was discovered theoretically by 't Hooft and Alexander Polyakov , flux tubes by Holger Bech Nielsen and Poul Olesen , and instantons by Polyakov and coauthors. These objects are inaccessible through perturbation theory. [ 8 ] : 4
Supersymmetry also appeared in the same period. The first supersymmetric QFT in four dimensions was built by Yuri Golfand and Evgeny Likhtman in 1970, but their result failed to garner widespread interest due to the Iron Curtain . Supersymmetry theories only took off in the theoretical community after the work of Julius Wess and Bruno Zumino in 1973, [ 8 ] : 7 but to date have not been widely accepted as part of the Standard Model due to lack of experimental evidence. [ 25 ]
Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of quantum gravity led to the development of string theory , [ 8 ] : 6 itself a type of two-dimensional QFT with conformal symmetry . [ 26 ] Joël Scherk and John Schwarz first proposed in 1974 that string theory could be the quantum theory of gravity. [ 27 ]
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to many-body systems in condensed matter physics .
Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of Yoichiro Nambu 's application of superconductor theory to elementary particles, while the concept of renormalization came out of the study of second-order phase transitions in matter. [ 28 ]
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle — phonons . Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems. [ 29 ]
Gauge theory is used to describe the quantization of magnetic flux in superconductors, the resistivity in the quantum Hall effect , as well as the relation between frequency and voltage in the AC Josephson effect . [ 29 ]
For simplicity, natural units are used in the following sections, in which the reduced Planck constant ħ and the speed of light c are both set to one.
A classical field is a function of spatial and time coordinates. [ 30 ] Examples include the gravitational field in Newtonian gravity g ( x , t ) and the electric field E ( x , t ) and magnetic field B ( x , t ) in classical electromagnetism . A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom . [ 30 ] [ 31 ]
Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles ( photons ), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.
Canonical quantization and path integrals are two common formulations of QFT. [ 32 ] : 61 To motivate the fundamentals of QFT, an overview of classical field theory follows.
The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as ϕ ( x , t ) , where x is the position vector, and t is the time. Suppose the Lagrangian of the field, L {\displaystyle L} , is
where L {\displaystyle {\mathcal {L}}} is the Lagrangian density, ϕ ˙ {\displaystyle {\dot {\phi }}} is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian: [ 1 ] : 16
we obtain the equations of motion for the field, which describe the way it varies in time and space:
This is known as the Klein–Gordon equation . [ 1 ] : 17
The Klein–Gordon equation is a wave equation , so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform ) as follows:
where a is a complex number (normalized by convention), * denotes complex conjugation , and ω p is the frequency of the normal mode:
Thus each normal mode corresponding to a single p can be seen as a classical harmonic oscillator with frequency ω p . [ 1 ] : 21,26
The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator .
The displacement of a classical harmonic oscillator is described by
where a is a complex number (normalized by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label x of a quantum field.
For a quantum harmonic oscillator, x ( t ) is promoted to a linear operator x ^ ( t ) {\displaystyle {\hat {x}}(t)} :
Complex numbers a and a * are replaced by the annihilation operator a ^ {\displaystyle {\hat {a}}} and the creation operator a ^ † {\displaystyle {\hat {a}}^{\dagger }} , respectively, where † denotes Hermitian conjugation . The commutation relation between the two is
The Hamiltonian of the simple harmonic oscillator can be written as
The vacuum state | 0 ⟩ {\displaystyle |0\rangle } , which is the lowest energy state, is defined by
and has energy 1 2 ℏ ω . {\displaystyle {\frac {1}{2}}\hbar \omega .} One can easily check that [ H ^ , a ^ † ] = ℏ ω a ^ † , {\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega {\hat {a}}^{\dagger },} which implies that a ^ † {\displaystyle {\hat {a}}^{\dagger }} increases the energy of the simple harmonic oscillator by ℏ ω {\displaystyle \hbar \omega } . For example, the state a ^ † | 0 ⟩ {\displaystyle {\hat {a}}^{\dagger }|0\rangle } is an eigenstate of energy 3 ℏ ω / 2 {\displaystyle 3\hbar \omega /2} .
Any energy eigenstate state of a single harmonic oscillator can be obtained from | 0 ⟩ {\displaystyle |0\rangle } by successively applying the creation operator a ^ † {\displaystyle {\hat {a}}^{\dagger }} : [ 1 ] : 20 and any state of the system can be expressed as a linear combination of the states
A similar procedure can be applied to the real scalar field ϕ , by promoting it to a quantum field operator ϕ ^ {\displaystyle {\hat {\phi }}} , while the annihilation operator a ^ p {\displaystyle {\hat {a}}_{\mathbf {p} }} , the creation operator a ^ p † {\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }} and the angular frequency ω p {\displaystyle \omega _{\mathbf {p} }} are now for a particular p :
Their commutation relations are: [ 1 ] : 21
where δ is the Dirac delta function . The vacuum state | 0 ⟩ {\displaystyle |0\rangle } is defined by
Any quantum state of the field can be obtained from | 0 ⟩ {\displaystyle |0\rangle } by successively applying creation operators a ^ p † {\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }} (or by a linear combination of such states), e.g. [ 1 ] : 22
While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space , which can account for the fact that particle numbers are not fixed in relativistic quantum systems. [ 33 ] The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization . [ 1 ] : 19
The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields , [ 1 ] : 52 vector fields ( e.g. the electromagnetic field), and even strings . [ 34 ] However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary.
The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field: [ 1 ] : 77
where μ is a spacetime index, ∂ 0 = ∂ / ∂ t , ∂ 1 = ∂ / ∂ x 1 {\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}} , etc. The summation over the index μ has been omitted following the Einstein notation . If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.
The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state | ϕ I ⟩ {\displaystyle |\phi _{I}\rangle } at time t = 0 to some final state | ϕ F ⟩ {\displaystyle |\phi _{F}\rangle } at t = T , the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian ( i.e. generator of time evolution ), then [ 32 ] : 10
Taking the limit N → ∞ , the above product of integrals becomes the Feynman path integral: [ 1 ] : 282 [ 32 ] : 12
where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H via Legendre transformation . The initial and final conditions of the path integral are respectively
In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.
In calculations, one often encounters expression like ⟨ 0 | T { ϕ ( x ) ϕ ( y ) } | 0 ⟩ or ⟨ Ω | T { ϕ ( x ) ϕ ( y ) } | Ω ⟩ {\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle } in the free or interacting theory, respectively. Here, x {\displaystyle x} and y {\displaystyle y} are position four-vectors , T {\displaystyle T} is the time ordering operator that shuffles its operands so the time-components x 0 {\displaystyle x^{0}} and y 0 {\displaystyle y^{0}} increase from right to left, and | Ω ⟩ {\displaystyle |\Omega \rangle } is the ground state (vacuum state) of the interacting theory, different from the free ground state | 0 ⟩ {\displaystyle |0\rangle } . This expression represents the probability amplitude for the field to propagate from y to x , and goes by multiple names, like the two-point propagator , two-point correlation function , two-point Green's function or two-point function for short. [ 1 ] : 82
The free two-point function, also known as the Feynman propagator , can be found for the real scalar field by either canonical quantization or path integrals to be [ 1 ] : 31,288 [ 32 ] : 23
In an interacting theory, where the Lagrangian or Hamiltonian contains terms L I ( t ) {\displaystyle L_{I}(t)} or H I ( t ) {\displaystyle H_{I}(t)} that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function.
In canonical quantization, the two-point correlation function can be written as: [ 1 ] : 87
where ε is an infinitesimal number and ϕ I is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in ϕ 4 {\displaystyle \phi ^{4}} -theory, the interacting term of the Hamiltonian is H I ( t ) = ∫ d 3 x λ 4 ! ϕ I ( x ) 4 {\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}} , [ 1 ] : 84 and the expansion of the two-point correlator in terms of λ {\displaystyle \lambda } becomes ⟨ Ω | T { ϕ ( x ) ϕ ( y ) } | Ω ⟩ = ∑ n = 0 ∞ ( − i λ ) n ( 4 ! ) n n ! ∫ d 4 z 1 ⋯ ∫ d 4 z n ⟨ 0 | T { ϕ I ( x ) ϕ I ( y ) ϕ I ( z 1 ) 4 ⋯ ϕ I ( z n ) 4 } | 0 ⟩ ∑ n = 0 ∞ ( − i λ ) n ( 4 ! ) n n ! ∫ d 4 z 1 ⋯ ∫ d 4 z n ⟨ 0 | T { ϕ I ( z 1 ) 4 ⋯ ϕ I ( z n ) 4 } | 0 ⟩ . {\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.} This perturbation expansion expresses the interacting two-point function in terms of quantities ⟨ 0 | ⋯ | 0 ⟩ {\displaystyle \langle 0|\cdots |0\rangle } that are evaluated in the free theory.
In the path integral formulation, the two-point correlation function can be written [ 1 ] : 284
where L {\displaystyle {\mathcal {L}}} is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in λ , reducing the interacting two-point function to quantities in the free theory.
Wick's theorem further reduce any n -point correlation function in the free theory to a sum of products of two-point correlation functions. For example,
Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory. [ 1 ] : 90 This makes the Feynman propagator one of the most important quantities in quantum field theory.
Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram . For example, the λ 1 term in the two-point correlation function in the ϕ 4 theory is
After applying Wick's theorem, one of the terms is
This term can instead be obtained from the Feynman diagram
The diagram consists of
Every vertex corresponds to a single ϕ {\displaystyle \phi } field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:
With the symmetry factor 2 {\displaystyle 2} , following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space. [ 1 ] : 91–94
In order to compute the n -point correlation function to the k -th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,
is equal to the sum of (expressions corresponding to) all connected diagrams with n external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the ϕ 4 interaction theory discussed above, every vertex must have four legs. [ 1 ] : 98
In realistic applications, the scattering amplitude of a certain interaction or the decay rate of a particle can be computed from the S-matrix , which itself can be found using the Feynman diagram method. [ 1 ] : 102–115
Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n -loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction. [ 32 ] : 44 Lines whose end points are vertices can be thought of as the propagation of virtual particles . [ 1 ] : 31
Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.
Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ , have no physical meaning — m , λ , and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ , obtain expressions for the physical quantities, and then take the limit Λ → ∞ . This is an example of regularization , a class of methods to treat divergences in QFT, with Λ being the regulator.
The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ 4 theory, the field strength is first redefined:
where ϕ is the bare field, ϕ r is the renormalized field, and Z is a constant to be determined. The Lagrangian density becomes:
where m r and λ r are the experimentally measurable, renormalized, mass and coupling constant, respectively, and
are constants to be determined. The first three terms are the ϕ 4 Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or dimensional regularization ); call the regulator Λ . Compute Feynman diagrams, in which divergent terms will depend on Λ . Then, define δ Z , δ m , and δ λ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained. [ 1 ] : 323–326
It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalizable QFT, [ 1 ] : 719–727 while quantum gravity is non-renormalizable. [ 1 ] : 798 [ 32 ] : 421
The renormalization group , developed by Kenneth Wilson , is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales. [ 1 ] : 393 The way in which each parameter changes with scale is described by its β function . [ 1 ] : 417 Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation . [ 1 ] : 410–411
As an example, the coupling constant in QED, namely the elementary charge e , has the following β function:
where Λ is the energy scale under which the measurement of e is performed. This differential equation implies that the observed elementary charge increases as the scale increases. [ 35 ] The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant. [ 1 ] : 420
The coupling constant g in quantum chromodynamics , a non-Abelian gauge theory based on the symmetry group SU(3) , has the following β function:
where N f is the number of quark flavours . In the case where N f ≤ 16 (the Standard Model has N f = 6 ), the coupling constant g decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom . [ 1 ] : 531
Conformal field theories (CFTs) are special QFTs that admit conformal symmetry . They are insensitive to changes in the scale, as all their coupling constants have vanishing β function. (The converse is not true, however — the vanishing of all β functions does not imply conformal symmetry of the theory.) [ 36 ] Examples include string theory [ 26 ] and N = 4 supersymmetric Yang–Mills theory . [ 37 ]
According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off Λ , i.e. that the theory is no longer valid at energies higher than Λ , and all degrees of freedom above the scale Λ are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable effective field theory . [ 1 ] : 402–403 The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them. [ 8 ] : 2 According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off Λ from calculations in such a theory merely indicates that new physical phenomena appear at scales above Λ , where a new theory is necessary. [ 32 ] : 156
The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and ϕ 4 theory of the real scalar field. A similar process can be done for other types of fields, including the complex scalar field, the vector field , and the Dirac field , as well as other types of interaction terms, including the electromagnetic interaction and the Yukawa interaction .
As an example, quantum electrodynamics contains a Dirac field ψ representing the electron field and a vector field A μ representing the electromagnetic field ( photon field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential , rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:
where γ μ are Dirac matrices , ψ ¯ = ψ † γ 0 {\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}} , and F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} is the electromagnetic field strength . The parameters in this theory are the (bare) electron mass m and the (bare) elementary charge e . The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories. [ 1 ] : 78
Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of electrons, while those pointing backward in time represent the propagation of positrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.
If the following transformation to the fields is performed at every spacetime point x (a local transformation), then the QED Lagrangian remains unchanged, or invariant:
where α ( x ) is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action ) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory. [ 1 ] : 482–483 Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations e i α ( x ) {\displaystyle e^{i\alpha (x)}} and e i α ′ ( x ) {\displaystyle e^{i\alpha '(x)}} is yet another symmetry transformation e i [ α ( x ) + α ′ ( x ) ] {\displaystyle e^{i[\alpha (x)+\alpha '(x)]}} . For any α ( x ) , e i α ( x ) {\displaystyle e^{i\alpha (x)}} is an element of the U(1) group, thus QED is said to have U(1) gauge symmetry. [ 1 ] : 496 The photon field A μ may be referred to as the U(1) gauge boson .
U(1) is an Abelian group , meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups , giving rise to non-Abelian gauge theories (also known as Yang–Mills theories). [ 1 ] : 489 Quantum chromodynamics , which describes the strong interaction, is a non-Abelian gauge theory with an SU(3) gauge symmetry. It contains three Dirac fields ψ i , i = 1,2,3 representing quark fields as well as eight vector fields A a,μ , a = 1,...,8 representing gluon fields, which are the SU(3) gauge bosons. [ 1 ] : 547 The QCD Lagrangian density is: [ 1 ] : 490–491
where D μ is the gauge covariant derivative :
where g is the coupling constant, t a are the eight generators of SU(3) in the fundamental representation ( 3×3 matrices),
and f abc are the structure constants of SU(3) . Repeated indices i , j , a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:
where U ( x ) is an element of SU(3) at every spacetime point x :
The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly . For instance, in the path integral formulation, despite the invariance of the Lagrangian density L [ ϕ , ∂ μ ϕ ] {\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]} under a certain local transformation of the fields, the measure ∫ D ϕ {\textstyle \int {\mathcal {D}}\phi } of the path integral may change. [ 32 ] : 243 For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1) , in which all anomalies exactly cancel. [ 1 ] : 705–707
The theoretical foundation of general relativity , the equivalence principle , can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group . [ 38 ]
Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding conservation law . [ 1 ] : 17–18 [ 32 ] : 73 For example, the U(1) symmetry of QED implies charge conservation . [ 39 ]
Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field A μ , being a four-vector , has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarization . The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing A μ can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description. [ 32 ] : 168
To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally. [ 1 ] : 512–515 A more rigorous generalization of the Faddeev–Popov procedure is given by BRST quantization . [ 1 ] : 517
Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it. [ 1 ] : 347
To illustrate the mechanism, consider a linear sigma model containing N real scalar fields, described by the Lagrangian density:
where μ and λ are real parameters. The theory admits an O( N ) global symmetry:
The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field ϕ 0 satisfying
Without loss of generality, let the ground state be in the N -th direction:
The original N fields can be rewritten as:
and the original Lagrangian density as:
where k = 1, ..., N − 1 . The original O( N ) global symmetry is no longer manifest, leaving only the subgroup O( N − 1) . The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken. [ 1 ] : 349–350
Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, O( N ) has N ( N − 1)/2 continuous symmetries (the dimension of its Lie algebra ), while O( N − 1) has ( N − 1)( N − 2)/2 . The number of broken symmetries is their difference, N − 1 , which corresponds to the N − 1 massless fields π k . [ 1 ] : 351
On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson. [ 1 ] : 743–744
In the QFT of ferromagnetism , spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures. [ 32 ] : 199 In the Standard Model of elementary particles, the W and Z bosons , which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson , a process called the Higgs mechanism . [ 1 ] : 690
All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesized the existence of a type of symmetry, called supersymmetry , that relates bosons and fermions. [ 1 ] : 795 [ 32 ] : 443
The Standard Model obeys Poincaré symmetry , whose generators are the spacetime translations P μ and the Lorentz transformations J μν . [ 40 ] : 58–60 In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators Q α , called supercharges , which themselves transform as Weyl fermions . [ 1 ] : 795 [ 32 ] : 444 The symmetry group generated by all these generators is known as the super-Poincaré group . In general there can be more than one set of supersymmetry generators, Q α I , I = 1, ..., N , which generate the corresponding N = 1 supersymmetry, N = 2 supersymmetry, and so on. [ 1 ] : 795 [ 32 ] : 450 Supersymmetry can also be constructed in other dimensions, [ 41 ] most notably in (1+1) dimensions for its application in superstring theory . [ 42 ]
The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group. [ 32 ] : 448 Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), N = 4 supersymmetric Yang–Mills theory , [ 32 ] : 450 and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa. [ 32 ] : 444
If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity . [ 43 ]
Supersymmetry is a potential solution to many current problems in physics. For example, the hierarchy problem of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the grand unified scale or the Planck scale —can be resolved by relating the Higgs field and its super-partner, the Higgsino . Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of dark matter . [ 1 ] : 796–797 [ 44 ]
Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments. [ 1 ] : 797 [ 32 ] : 443
The ϕ 4 theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime.
In condensed matter physics , QFT is used to describe (2+1)-dimensional electron gases . [ 45 ] In high-energy physics , string theory is a type of (1+1)-dimensional QFT, [ 32 ] : 452 [ 26 ] while Kaluza–Klein theory uses gravity in extra dimensions to produce gauge theories in lower dimensions. [ 32 ] : 428–429
In Minkowski space, the flat metric η μν is used to raise and lower spacetime indices in the Lagrangian, e.g.
where η μν is the inverse of η μν satisfying η μρ η ρν = δ μ ν .
For QFTs in curved spacetime on the other hand, a general metric (such as the Schwarzschild metric describing a black hole ) is used:
where g μν is the inverse of g μν .
For a real scalar field, the Lagrangian density in a general spacetime background is
where g = det( g μν ) , and ∇ μ denotes the covariant derivative . [ 46 ] The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.
The correlation functions and physical predictions of a QFT depend on the spacetime metric g μν . For a special class of QFTs called topological quantum field theories (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric. [ 47 ] : 36 QFTs in curved spacetime generally change according to the geometry (local structure) of the spacetime background, while TQFTs are invariant under spacetime diffeomorphisms but are sensitive to the topology (global structure) of spacetime. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime. Chern–Simons theory is an example of TQFT and has been used to construct models of quantum gravity. [ 48 ] Applications of TQFT include the fractional quantum Hall effect and topological quantum computers . [ 49 ] : 1–5 The world line trajectory of fractionalized particles (known as anyons ) can form a link configuration in the spacetime, [ 50 ] which relates the braiding statistics of anyons in physics to the
link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond. [ 51 ]
Using perturbation theory , the total effect of a small interaction term can be approximated order by order by a series expansion in the number of virtual particles participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a Feynman diagram . The electromagnetic force between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the W and Z bosons carry the weak interaction, while gluons carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as 't Hooft–Polyakov monopole , domain wall , flux tube , and instanton . [ 8 ] Examples of QFTs that are completely solvable non-perturbatively include minimal models of conformal field theory [ 52 ] and the Thirring model . [ 53 ]
In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem , there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined. [ 54 ]
However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, Kevin Costello 's monograph Renormalization and Effective Field Theory [ 55 ] provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of Kadanoff , Wilson , and Polchinski , together with the Batalin-Vilkovisky approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory, [ 56 ] can be given a sound mathematical interpretation from their finite-dimensional analogues. [ 57 ]
Since the 1950s, [ 58 ] theoretical physicists and mathematicians have attempted to organize all QFTs into a set of axioms , in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory , a subfield of mathematical physics , [ 59 ] : 2 which has led to such results as CPT theorem , spin–statistics theorem , and Goldstone's theorem , [ 58 ] and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions, [ 60 ] the three-dimensional scalar field theories with a quartic interaction, etc. [ 61 ]
Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms . [ 62 ]
Algebraic quantum field theory is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag–Kastler axioms . [ 59 ] : 2–3 One way to construct theories satisfying Wightman axioms is to use Osterwalder–Schrader axioms , which give the necessary and sufficient conditions for a real time theory to be obtained from an imaginary time theory by analytic continuation ( Wick rotation ). [ 59 ] : 10
Yang–Mills existence and mass gap , one of the Millennium Prize Problems , concerns the well-defined existence of Yang–Mills theories as set out by the above axioms. The full problem statement is as follows. [ 63 ]
Prove that for any compact simple gauge group G , a non-trivial quantum Yang–Mills theory exists on R 4 {\displaystyle \mathbb {R} ^{4}} and has a mass gap Δ > 0 . Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964) , Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975) . | https://en.wikipedia.org/wiki/Quantum_field_theory |
Quantum finance is an interdisciplinary research field, applying theories and methods developed by quantum physicists and economists in order to solve problems in finance . It is a branch of econophysics .
Most quantum option pricing research typically focuses on the quantization of the classical Black–Scholes–Merton equation from the perspective of continuous equations like the Schrödinger equation . Emmanuel Haven builds on the work of Zeqian Chen and others, [ 1 ] but considers the market from the perspective of the Schrödinger equation . [ 2 ] The key message in Haven's work is that the Black–Scholes–Merton equation is really a special case of the Schrödinger equation where markets are assumed to be efficient. The Schrödinger-based equation that Haven derives has a parameter ħ (not to be confused with the complex conjugate of h ) that represents the amount of arbitrage that is present in the market resulting from a variety of sources including non-infinitely fast price changes, non-infinitely fast information dissemination and unequal wealth among traders. Haven argues that by setting this value appropriately, a more accurate option price can be derived, because in reality, markets are not truly efficient.
This is one of the reasons why it is possible that a quantum option pricing model could be more accurate than a classical one. Belal E. Baaquie has published many papers on quantum finance and even written a book that brings many of them together. [ 3 ] [ 4 ] Core to Baaquie's research and others like Matacz are Richard Feynman 's path integrals . [ 5 ]
Baaquie applies path integrals to several exotic options and presents analytical results comparing his results to the results of Black–Scholes–Merton equation showing that they are very similar. Edward Piotrowski et al. take a different approach by changing the Black–Scholes–Merton assumption regarding the behavior of the stock underlying the option. [ 6 ] Instead of assuming it follows a Wiener–Bachelier process , [ 7 ] they assume that it follows an Ornstein–Uhlenbeck process . [ 8 ] With this new assumption in place, they derive a quantum finance model as well as a European call option formula.
Other models such as Hull–White and Cox–Ingersoll–Ross have successfully used the same approach in the classical setting with interest rate derivatives. [ 9 ] [ 10 ] Andrei Khrennikov builds on the work of Haven and others and further bolsters the idea that the market efficiency assumption made by the Black–Scholes–Merton equation may not be appropriate. [ 11 ] To support this idea, Khrennikov builds on a framework of contextual probabilities using agents as a way of overcoming criticism of applying quantum theory to finance. Luigi Accardi and Andreas Boukas again quantize the Black–Scholes–Merton equation, but in this case, they also consider the underlying stock to have both Brownian and Poisson processes. [ 12 ]
Chen published a paper in 2001, [ 1 ] where he presents a quantum binomial options pricing model or simply abbreviated as the quantum binomial model. Metaphorically speaking, Chen's quantum binomial options pricing model (referred to
hereafter as the quantum binomial model) is to existing quantum finance models what the Cox–Ross–Rubinstein classical binomial options pricing model was to the Black–Scholes–Merton model: a discretized and simpler version of the same result. These simplifications make the respective theories not only easier to analyze but also easier to implement on a computer.
In the multi-step model the quantum pricing formula is:
which is the equivalent of the Cox–Ross–Rubinstein binomial options pricing model formula as follows:
This shows that assuming stocks behave according to Maxwell–Boltzmann statistics , the quantum binomial model does indeed collapse to the classical binomial model.
Quantum volatility is as follows as per Keith Meyer: [ 13 ]
Maxwell–Boltzmann statistics can be replaced by the quantum Bose–Einstein statistics resulting in the following option price formula:
The Bose–Einstein equation will produce option prices that will differ from those produced by the Cox–Ross–Rubinstein option
pricing formula in certain circumstances. This is because the stock is being treated like a quantum boson particle instead of a classical particle.
Patrick Rebentrost showed in 2018 that an algorithm exists for quantum computers capable of pricing financial derivatives with a square root advantage over classical methods. [ 14 ] This development marks a shift from using quantum mechanics to gain insight into functional finance, to using quantum systems- quantum computers, to perform those calculations.
In 2020 David Orrell proposed an option-pricing model based on a quantum walk which can run on a photonics device. [ 15 ] [ 16 ] [ 17 ]
In their review of Baaquie's work, Arioli and Valente point out that, unlike Schrödinger's equation, the Black-Scholes-Merton equation uses no imaginary numbers. Since quantum characteristics in physics like superposition and entanglement are a result of the imaginary numbers, Baaquie's numerical success must result from effects other than quantum ones. [ 18 ] : 668 Rickles critiques Baaquies's work on economics grounds: empirical economic data are not random so they don't need a quantum randomness explanation. [ 19 ] : 969 | https://en.wikipedia.org/wiki/Quantum_finance |
In quantum physics , a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation ) is the temporary random change in the amount of energy in a point in space , [ 2 ] as prescribed by Werner Heisenberg 's uncertainty principle . They are minute random fluctuations in the values of the fields which represent elementary particles, such as electric and magnetic fields which represent the electromagnetic force carried by photons , W and Z fields which carry the weak force , and gluon fields which carry the strong force . [ 3 ]
The uncertainty principle states the uncertainty in energy and time can be related by [ 4 ] Δ E Δ t ≥ 1 2 ℏ {\displaystyle \Delta E\,\Delta t\geq {\tfrac {1}{2}}\hbar ~} , where 1 / 2 ħ ≈ 5.272 86 × 10 −35 J⋅s . This means that pairs of virtual particles with energy Δ E {\displaystyle \Delta E} and lifetime shorter than Δ t {\displaystyle \Delta t} are continually created and annihilated in empty space. Although the particles are not directly detectable, the cumulative effects of these particles are measurable. For example, without quantum fluctuations, the "bare" mass and charge of elementary particles would be infinite; from renormalization theory the shielding effect of the cloud of virtual particles is responsible for the finite mass and charge of elementary particles.
Another consequence is the Casimir effect . One of the first observations which was evidence for vacuum fluctuations was the Lamb shift in hydrogen. In July 2020, scientists reported that quantum vacuum fluctuations can influence the motion of macroscopic, human-scale objects by measuring correlations below the standard quantum limit between the position/momentum uncertainty of the mirrors of LIGO and the photon number/phase uncertainty of light that they reflect. [ 5 ] [ 6 ] [ 7 ]
In quantum field theory , fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations of a quantum field (at least for a free field; for interacting fields, renormalization substantially complicates matters). An illustration of this distinction can be seen by considering quantum and classical Klein–Gordon fields: [ 8 ] For the quantized Klein–Gordon field in the vacuum state , we can calculate the probability density that we would observe a configuration φ t ( x ) {\displaystyle \varphi _{t}(x)} at a time t in terms of its Fourier transform φ ~ t ( k ) {\displaystyle {\tilde {\varphi }}_{t}(k)} to be
In contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration φ t ( x ) {\displaystyle \varphi _{t}(x)} at a time t {\displaystyle t} is
These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by the Planck constant ℏ {\displaystyle \hbar } , just as the amplitude of thermal fluctuations is controlled by k B T {\displaystyle k_{\text{B}}T} , where k B is the Boltzmann constant . Note that the following three points are closely related:
A classical continuous random field can be constructed that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory ( measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible – in quantum-mechanical terms they always commute). | https://en.wikipedia.org/wiki/Quantum_fluctuation |
A quantum fluid refers to any system that exhibits quantum mechanical effects at the macroscopic level such as superfluids , superconductors , ultracold atoms , etc. Typically, quantum fluids arise in situations where both quantum mechanical effects and quantum statistical effects are significant.
Most matter is either solid or gaseous (at low densities) near absolute zero . However, for the cases of helium-4 and its isotope helium-3 , there is a pressure range where they can remain liquid down to absolute zero because the amplitude of the quantum fluctuations experienced by the helium atoms is larger than the inter-atomic distances.
In the case of solid quantum fluids, it is only a fraction of its electrons or protons that behave like a “fluid”. One prominent example is that of superconductivity where quasi-particles made up of pairs of electrons and a phonon act as bosons which are then capable of collapsing into the ground state to establish a supercurrent with a resistivity near zero.
Quantum mechanical effects become significant for physics in the range of the de Broglie wavelength . For condensed matter, this is when the de Broglie wavelength of a particle is greater than the spacing between the particles in the lattice that comprises the matter.
The de Broglie wavelength associated with a massive particle is
where h is the Planck constant. The momentum can be found from the kinetic theory of gases , where
Here, the temperature can be found as
Of course, we can replace the momentum here with the momentum derived from the de Broglie wavelength like so:
Hence, we can say that quantum fluids will manifest at approximate temperature regions where λ > d {\displaystyle \lambda >d} , where d is the lattice spacing (or inter-particle spacing). Mathematically, this is stated like so:
It is easy to see how the above definition relates to the particle density, n. We can write
as n = 1 d 3 {\displaystyle n={\frac {1}{d^{3}}}} for a three dimensional lattice
The above temperature limit T {\displaystyle T} has different meaning depending on the quantum statistics followed by each system, but generally refers to the point at which the system manifests quantum fluid properties. For a system of fermions , T {\displaystyle T} is an estimation of the Fermi energy of the system, where processes important to phenomena such as superconductivity take place. For bosons , T {\displaystyle T} gives an estimation of the Bose-Einstein condensation temperature. | https://en.wikipedia.org/wiki/Quantum_fluid |
A Quantum Flux Parametron ( QFP ) is a digital logic implementation technology based on superconducting Josephson junctions . [ 1 ] QFP's were invented by Eiichi Goto at the University of Tokyo as an improvement over his earlier parametron based digital logic technology, which did not use superconductivity effects or Josephson junctions. The Josephson junctions on QFP integrated circuits to improve speed and energy efficiency enormously over the parametrons.
In some applications, the complexity of the cryogenic cooling system required is negligible compared to the potential speed gains. While his design makes use of quantum principles, it is not a quantum computer technology, gaining speed only through higher clock speeds.
Apart from the speed advantage over traditional CMOS integrated circuit design is that parametrons can be operated with zero energy loss (no local increase in entropy ), making reversible computing possible. Low energy use and heat generation is critical in supercomputer design, where thermal load per unit volume has become one of the main limiting factors.
A related technology is the Rapid Single Flux Quantum digital logic.
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_flux_parametron |
Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory , reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity , the defining axioms of quantum theory are quite ad hoc , with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.
There exist different approaches to resolve this conceptual gap:
Research in quantum foundations is structured along these roads.
Two or more separate parties conducting measurements over a quantum state can observe correlations which cannot be explained with any local hidden variable theory . [ 1 ] [ 2 ] Whether this should be regarded as proving that the physical world itself is "nonlocal" is a topic of debate, [ 3 ] [ 4 ] but the terminology of "quantum nonlocality" is commonplace. Nonlocality research efforts in quantum foundations focus on determining the exact limits that classical or quantum physics enforces on the correlations observed in a Bell experiment or more complex causal scenarios. [ 5 ] This research program has so far provided a generalization of Bell's theorem that allows falsifying all classical theories with a superluminal, yet finite, hidden influence. [ 6 ]
Nonlocality can be understood as an instance of quantum contextuality . A situation is contextual when the value of an observable depends on the context in which it is measured (namely, on which other observables are being measured as well). The original definition of measurement contextuality can be extended to state preparations and even general physical transformations. [ 7 ]
A physical property is epistemic when it represents our knowledge or beliefs on the value of a second, more fundamental feature. The probability of an event to occur is an example of an epistemic property. In contrast, a non-epistemic or ontic variable captures the notion of a “real” property of the system under consideration.
There is an on-going debate on whether the wave-function represents the epistemic state of a yet to be discovered ontic variable or, on the contrary, it is a fundamental entity. [ 8 ] Under some physical assumptions, the Pusey–Barrett–Rudolph (PBR) theorem demonstrates the inconsistency of quantum states as epistemic states, in the sense above. [ 9 ] Note that, in QBism [ 10 ] and Copenhagen -type [ 11 ] views, quantum states are still regarded as epistemic, not with respect to some ontic variable, but to one's expectations about future experimental outcomes. The PBR theorem does not exclude such epistemic views on quantum states.
Some of the counter-intuitive aspects of quantum theory, as well as the difficulty to extend it, follow from the fact that its defining axioms lack a physical motivation. An active area of research in quantum foundations is therefore to find alternative formulations of quantum theory which rely on physically compelling principles. Those efforts come in two flavors, depending on the desired level of description of the theory: the so-called Generalized Probabilistic Theories approach and the Black boxes approach.
Generalized Probabilistic Theories (GPTs) are a general framework to describe the operational features of arbitrary physical theories. Essentially, they provide a statistical description of any experiment combining state preparations, transformations and measurements. The framework of GPTs can accommodate classical and quantum physics, as well as hypothetical non-quantum physical theories which nonetheless possess quantum theory's most remarkable features, such as entanglement or teleportation. [ 12 ] Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory. [ 13 ]
L. Hardy introduced the concept of GPT in 2001, in an attempt to re-derive quantum theory from basic physical principles. [ 13 ] Although Hardy's work was very influential (see the follow-ups below), one of his axioms was regarded as unsatisfactory: it stipulated that, of all the physical theories compatible with the rest of the axioms, one should choose the simplest one. [ 14 ] The work of Dakic and Brukner eliminated this “axiom of simplicity” and provided a reconstruction of quantum theory based on three physical principles. [ 14 ] This was followed by the more rigorous reconstruction of Masanes and Müller. [ 15 ]
Axioms common to these three reconstructions are:
An alternative GPT reconstruction proposed by Chiribella, D'Ariano and Perinotti [ 16 ] [ 17 ] around the same time is also based on the
The use of purification to characterize quantum theory has been criticized on the grounds that it also applies in the Spekkens toy model . [ 18 ]
To the success of the GPT approach, it can be countered that all such works just recover finite dimensional quantum theory. In addition, none of the previous axioms can be experimentally falsified unless the measurement apparatuses are assumed to be tomographically complete .
Categorical Quantum Mechanics (CQM) or Process Theories are a general framework to describe physical theories, with an emphasis on processes and their compositions. [ 19 ] It was pioneered by Samson Abramsky and Bob Coecke . Besides its influence in quantum foundations, most notably the use of a diagrammatic formalism, CQM also plays an important role in quantum technologies, most notably in the form of ZX-calculus . It also has been used to model theories outside of physics, for example the DisCoCat compositional natural language meaning model.
In the black box or device-independent framework, an experiment is regarded as a black box where the experimentalist introduces an input (the type of experiment) and obtains an output (the outcome of the experiment). Experiments conducted by two or more parties in separate labs are hence described by their statistical correlations alone.
From Bell's theorem , we know that classical and quantum physics predict different sets of allowed correlations. It is expected, therefore, that far-from-quantum physical theories should predict correlations beyond the quantum set. In fact, there exist instances of theoretical non-quantum correlations which, a priori, do not seem physically implausible. [ 20 ] [ 21 ] [ 22 ] The aim of device-independent reconstructions is to show that all such supra-quantum examples are precluded by a reasonable physical principle.
The physical principles proposed so far include no-signalling, [ 22 ] Non-Trivial Communication Complexity, [ 23 ] No-Advantage for Nonlocal computation, [ 24 ] Information Causality , [ 25 ] Macroscopic Locality, [ 26 ] and Local Orthogonality. [ 27 ] All these principles limit the set of possible correlations in non-trivial ways. Moreover, they are all device-independent: this means that they can be falsified under the assumption that we can decide if two or more events are space-like separated. The drawback of the device-independent approach is that, even when taken together, all the afore-mentioned physical principles do not suffice to single out the set of quantum correlations. [ 28 ] In other words: all such reconstructions are partial.
An interpretation of quantum theory is a correspondence between the elements of its mathematical formalism and physical phenomena. For instance, in the pilot wave theory , the quantum wave function is interpreted as a field that guides the particle trajectory and evolves with it via a system of coupled differential equations. Most interpretations of quantum theory stem from the desire to solve the quantum measurement problem .
In an attempt to reconcile quantum and classical physics, or to identify non-classical models with a dynamical causal structure, some modifications of quantum theory have been proposed.
Collapse models posit the existence of natural processes which periodically localize the wave-function. [ 29 ] Such theories provide an explanation to the nonexistence of superpositions of macroscopic objects, at the cost of abandoning unitarity and exact energy conservation .
In Sorkin 's quantum measure theory (QMT), physical systems are not modeled via unitary rays and Hermitian operators, but through a single matrix-like object, the decoherence functional. [ 30 ] The entries of the decoherence functional determine the feasibility to experimentally discriminate between two or more different sets of classical histories, as well as the probabilities of each experimental outcome. In some models of QMT the decoherence functional is further constrained to be positive semidefinite (strong positivity). Even under the assumption of strong positivity, there exist models of QMT which generate stronger-than-quantum Bell correlations. [ 31 ]
The formalism of process matrices starts from the observation that, given the structure of quantum states, the set of feasible quantum operations follows from positivity considerations. Namely, for any linear map from states to probabilities one can find a physical system where this map corresponds to a physical measurement. Likewise, any linear transformation that maps composite states to states corresponds to a valid operation in some physical system. In view of this trend, it is reasonable to postulate that any high-order map from quantum instruments (namely, measurement processes) to probabilities should also be physically realizable. [ 32 ] Any such map is termed a process matrix. As shown by Oreshkov et al., [ 32 ] some process matrices describe situations where the notion of global causality breaks.
The starting point of this claim is the following mental experiment: two parties, Alice and Bob , enter a building and end up in separate rooms. The rooms have ingoing and outgoing channels from which a quantum system periodically enters and leaves the room. While those systems are in the lab, Alice and Bob are able to interact with them in any way; in particular, they can measure some of their properties.
Since Alice and Bob's interactions can be modeled by quantum instruments, the statistics they observe when they apply one instrument or another are given by a process matrix. As it turns out, there exist process matrices which would guarantee that the measurement statistics collected by Alice and Bob is incompatible with Alice interacting with her system at the same time, before or after Bob, or any convex combination of these three situations. [ 32 ] Such processes are called acausal. | https://en.wikipedia.org/wiki/Quantum_foundations |
Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
This theory is based on the physics of information much like quantum computing .
In 1969, John Clauser , Michael Horne , Abner Shimony , and Richard Holt (often referred to collectively as "CHSH") wrote an often-cited paper describing experiments which could be used to prove Bell's theorem . In one part of this paper, they describe a game where a player could have a better chance of winning by using quantum strategies than would be possible classically. While game theory was not explicitly mentioned in this paper, it is an early outline of how quantum entanglement could be used to alter a game.
In 1999, a professor in the math department at the University of California at San Diego named David A. Meyer first published Quantum Strategies which details a quantum version of the classical game theory game, matching pennies . [ 1 ] In the quantum version, players are allowed access to quantum signals through the phenomenon of quantum entanglement. [ 2 ] In the same year, Jens Eisert , Martin Wilkens and Maciej Lewenstein published work entitled Quantum Games and Quantum Strategies [ 3 ] that explored the role of quantum strategies in canonical two-player games such as the prisoner's dilemma .
Since Meyer's paper on the one hand and the
Eisert-Wilkens-Lewenstein paper on the other, a large number of publications have been published exploring quantum games and the way that quantum strategies could be used in games that have been commonly studied in classical game theory .
The information transfer that occurs during a game can be viewed as a physical process.
In the simplest case of a classical game between two players with two strategies each, both the players can use a bit (a '0' or a '1') to convey their choice of strategy. A popular example of such a game is the prisoners' dilemma , where each of the convicts can either cooperate or defect : withholding knowledge or revealing that the other committed the crime. In the quantum version of the game, the bit is replaced by the qubit , which is a quantum superposition of two or more base states. In the case of a two-strategy game this can be physically implemented by the use of an entity like the electron which has a superposed spin state, with the base states being +1/2 (plus half) and −1/2 (minus half). Each of the spin states can be used to represent each of the two strategies available to the players. When a measurement is made on the electron, it collapses to one of the base states, thus conveying the strategy used by the player.
The set of qubits which are initially provided to each of the players (to be used to convey their choice of strategy) may be entangled. For instance, an entangled pair of qubits implies that an operation performed on one of the qubits, affects the other qubit as well, thus altering the expected pay-offs of the game. A simple example of this is a quantum version [ 4 ] of the Two-up coin game in which the coins are entangled.
The job of a player in a game is to choose a strategy. In terms of bits this means that the player has to choose between 'flipping' the bit to its opposite state or leaving its current state untouched. When extended to the quantum domain this implies that the player can rotate the qubit to a new state, thus changing the probability amplitudes of each of the base states. Such operations on the qubits are required to be unitary transformations on the initial state of the qubit. This is different from the classical procedure which chooses the strategies with some statistical probabilities.
Introducing quantum information into multiplayer games allows a new type of "equilibrium strategy" which is not found in traditional games. The entanglement of players' choices can have the effect of a contract by preventing players from profiting from other player's betrayal . [ 5 ]
Quantum Prisoner's Dilemma
The classical Prisoner's Dilemma is a game played between two players with a choice to cooperate with or betray their opponent. Classically, the dominant strategy is to always choose betrayal. When both players choose this strategy every turn, they each ensure a suboptimal profit, but cannot lose, and the game is said to have reached a Nash equilibrium . Profit would be maximized for both players if each chose to cooperate every turn, but this is not the rational choice, thus a suboptimal solution is the dominant outcome. In the quantum Prisoner's Dilemma, both parties choosing to betray each other is still an equilibrium, however, there can also exist multiple Nash equilibriums that vary based on the entanglement of the initial states. In the case where the states are only slightly entangled, there exists a certain unitary operation for Alice so that if Bob chooses betrayal every turn, Alice will actually gain more profit than Bob and vice versa. Thus, a profitable equilibrium can be reached in 2 additional ways. The case where the initial state is most entangled shows the most change from the classical game. In this version of the game, Alice and Bob each have an operator Q that allows for a payout equal to mutual cooperation with no risk of betrayal. This is a Nash equilibrium that also happens to be Pareto optimal . [ 6 ]
Additionally, the quantum version of the Prisoner's Dilemma differs greatly from the classical version when the game is of unknown or infinite length. Classically, the infinite Prisoner's Dilemma has no defined fixed strategy but in the quantum version it is possible to develop an equilibrium strategy. [ 7 ]
Quantum Volunteer's Dilemma
The Volunteer's Dilemma is a well-known game in game theory that models the conflict players face when deciding whether to volunteer for a collective benefit, knowing that volunteering incurs a personal cost. One significant volunteer’s dilemma variant was introduced by Weesie and Franzen in 1998, [ 8 ] involves cost-sharing among volunteers. In this variant of the Volunteer's Dilemma, if there is no volunteer, all players receive a payoff of 0. If there is at least one volunteer, the reward of b units is distributed to all players. In contrast, the total cost of c units incurred by volunteering is divided equally among all the volunteers. It is shown that for classical mixed strategies setting, there is a unique symmetric Nash equilibrium and the Nash equilibrium is obtained by setting the probability of volunteering for each player to be the unique root in the open interval (0,1) of the degree-n polynomial g n {\displaystyle g_{n}} given by
g n ( α ) = ( 1 − α ) n − 1 ( 2 n α + 1 − α ) − 1. {\displaystyle g_{n}(\alpha )=(1-\alpha )^{n-1}(2n\alpha +1-\alpha )-1.}
In 2024, a quantum variant of the classical volunteer’s dilemma is introduced with b=2 and c=1 is studied, generalizing the classical setting by allowing players to utilize quantum strategies. [ 9 ] This is achieved by employing the Eisert–Wilkens–Lewenstein quantization framework. In this setting, the players received an entangled n-qubit state with each player controlling one qubit. The decision of each player can be viewed as determining two angles. Symmetric Nash equilibria that attain a payoff value of 2 − 1 / n {\displaystyle 2-1/n} for each player is shown and each player volunteers at this Nash Equilibrium. Furthermore, these Nash Equilibrium are Pareto optimal. It is shown that the payoff function of Nash equilibrium in the quantum setting is higher than the payoff of Nash equilibrium in the classical setting.
Quantum Card Game
A classically unfair card game can be played as follows: [ 10 ] There are two players, Alice and Bob. Alice has three cards: one has a star on both sides, one has a diamond on both sides, and one has a star on one side and a diamond on the other side. Alice places the three cards in a box and shakes it up, then Bob draws a card so that both players can only see one side of the card. If the card has the same markings on both sides, Alice wins. But if the card has different markings on each side, Bob wins. Clearly, this is an unfair game, where Alice has a probability of winning of 2/3 and Bob has a probability of winning of 1/3. Alice gives Bob one chance to "operate" on the box and then allows him to withdraw from the game if he would like, but he can only classically obtain information on one card from this operation, so the game is still unfair.
However, Alice and Bob can play a version of this game adjusted to allow for quantum strategies. If we describe the state of a card with a diamond facing up as | 0 ⟩ {\displaystyle |0\rangle } and the state where the star is facing up as | 1 ⟩ {\displaystyle |1\rangle } , after shaking the box up, we can describe the state of the face-up part of the cards as:
| r ⟩ = | r 0 r 1 r 2 ⟩ {\displaystyle |r\rangle =|r_{0}\ r_{1}\ r_{2}\rangle }
where each r k {\displaystyle r_{k}} is either 0 or 1.
Now, Bob can take advantage of his ability to operate on the box by constructing a machine as follows: First, he has a unitary matrix defined as U k = [ 1 0 0 e i π r k ] {\displaystyle U_{k}={\begin{bmatrix}1&0\\0&e^{i\pi r_{k}}\end{bmatrix}}} . This matrix is equal to I {\displaystyle I} if r k {\displaystyle r_{k}} is 0 and Z {\displaystyle Z} if r k {\displaystyle r_{k}} is 1. He then creates his machine by putting this matrix between two Hadamard gates, so his machine now looks as
H U k H = 1 2 [ 1 1 1 − 1 ] [ 1 0 0 e i π r k ] [ 1 1 1 − 1 ] = 1 2 [ 1 + e i π r k 1 − e i π r k 1 − e i π r k 1 + e i π r k ] . {\displaystyle HU_{k}H={\frac {1}{2}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}1&0\\0&e^{i\pi r_{k}}\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+e^{i\pi r_{k}}&1-e^{i\pi r_{k}}\\1-e^{i\pi r_{k}}&1+e^{i\pi r_{k}}\end{bmatrix}}.}
This machine operating on the state | 0 ⟩ {\displaystyle |0\rangle } gives
H U k H | 0 ⟩ = 1 + e i π r k 2 | 0 ⟩ + 1 − e i π r k 2 | 1 ⟩ = | r k ⟩ . {\displaystyle HU_{k}H|0\rangle ={\frac {1+e^{i\pi r_{k}}}{2}}|0\rangle +{\frac {1-e^{i\pi r_{k}}}{2}}|1\rangle =|r_{k}\rangle .}
So if Bob inputs | 000 ⟩ {\displaystyle |000\rangle } to his machine, he obtains
( H U k H ⊗ H U k H ⊗ H U k H ) | 000 ⟩ = | r 0 r 1 r 2 ⟩ {\displaystyle (HU_{k}H\otimes HU_{k}H\otimes HU_{k}H)|000\rangle =|r_{0}\ r_{1}\ r_{2}\rangle }
and he knows the state (i.e. the mark facing up) of all three of the cards. From here, Bob can draw one card, and then choose to either withdraw, or keep playing the game. Based on the first card that he draws, he can know from his knowledge of the face-up values of the cards whether or not he has drawn a card that will give him even chances of winning going forward (in which case he can continue to play a fair game) or if he has drawn the card that will guarantee that he loses the game. In this way, he can make the game fair for himself.
This is an example of a game where a quantum strategy can make a game fair for one player when it would be unfair for them with classical strategies.
Quantum Chess
Quantum Chess was first developed by a graduate student at the University of Southern California named Chris Cantwell. His motivation to develop the game was to expose non-physicists to the world of quantum mechanics. [ 11 ]
The game uses the same pieces as classical chess (8 pawns, 2 knights, 2 bishops, 2 rooks, 1 queen, 1 king) and is won in the same manner (by capturing the opponent's king). However, the pieces are allowed to obey laws of quantum mechanics such as superposition. By allowed the introduction of superposition, it becomes possible for pieces to occupy more than one square in an instance. The movement rules for each piece are the same as classical chess.
The biggest difference between quantum chess and classical chess is the check rule. Check is not included in quantum chess because it is possible for the king, as well as all other pieces, to occupy multiple spots on the grid at once. Another difference is the concept of movement to occupied space. Superposition also allows two occupies to share space or move through each other.
Capturing an opponent's piece is also slightly different in quantum chess than in classical chess. Quantum chess uses quantum measurement as a method of capturing. When attempting to capture an opponent's piece, a measurement is made to determine the probability of whether or not the space is occupied and if the path is blocked. If the probability is favorable, a move can be made to capture. [ 12 ]
PQ Penny Flip Game
The PQ penny flip game involves two players: Captain Picard and Q. Q places a penny in a box, then they take turns (Q, then Picard, then Q) either flipping or not flipping the penny without revealing its state to either player. After these three moves have been made, Q wins if the penny is heads up, and Picard if the penny is face down.
The classical Nash equilibrium has both players taking a mixed strategy with each move having a 50% chance of either flipping or not flipping the penny, and Picard and Q will each win the game 50% of the time using classical strategies.
Allowing for Q to use quantum strategies, namely applying a Hadamard gate to the state of the penny places it into a superposition of face up and down, represented by the quantum state
| ψ 1 ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) . {\displaystyle |\psi _{1}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ).}
In this state, if Picard does not flip the gate, then the state remains unchanged, and flipping the penny puts it into the state
| ψ 2 ⟩ = 1 2 ( | 1 ⟩ + | 0 ⟩ ) = | ψ 1 ⟩ . {\displaystyle |\psi _{2}\rangle ={\frac {1}{\sqrt {2}}}(|1\rangle +|0\rangle )=|\psi _{1}\rangle .}
Then, no matter Picard's move, Q can once again apply a Hadamard gate to the superposition which results in the penny being face up. In this way the quantization of Q's strategy guarantees a win against a player constrained by classical strategies.
This game is exemplary of how applying quantum strategies to classical games can shift an otherwise fair game in favor of the player using quantum strategies. [ 10 ]
The concepts of a quantum player, a zero-sum quantum game and the associated expected payoff were defined by A. Boukas in 1999 (for finite games) and in 2020 by L. Accardi and A. Boukas (for infinite games) within the framework of the spectral theorem for self-adjoint operators on Hilbert spaces. Quantum versions of Von Neumann's minimax theorem were proved. [ 13 ] [ 14 ]
Quantum game theory also offers a solution to Newcomb's Paradox .
Take the two boxes offered in Newcomb's game to be coupled, as the contents of box 2 depend on if the ignorant player takes box 1. Quantum game theory enables a situation such that foreknowledge by otherwise omniscient player isn't required in order to achieve the situation. If the otherwise omniscient player operates on the state of the two boxes using a Hadamard gate, then sets up a device that operates on the state defined by the two boxes to operate again using a Hadamard gate after the ignorant player's choice. Then, no matter the pure or mixed strategy that the ignorant player uses, the ignorant player's choice will lead to its corresponding outcome as defined by the premise of the game. Because choosing a strategy for the game, then changing it to fool to otherwise omniscient player (corresponding to operating on the game state using a NOT gate) cannot give the ignorant player an additional advantage, as the two Hadamard operations ensure that the only two outcomes are those defined by the chosen strategy. In this way, the expected situation is achieved no matter the ignorant player's strategy without requiring a system knowledgeable about that player's future. [ 15 ] | https://en.wikipedia.org/wiki/Quantum_game_theory |
In theoretical physics , quantum geometry is the set of mathematical concepts that generalize geometry to describe physical phenomena at distance scales comparable to the Planck length . At such distances, quantum mechanics has a profound effect on physical phenomena.
Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion. String theory , a leading candidate for a quantum theory of gravity, uses it to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry , topology -changing transitions [ clarification needed ] , minimal possible distance scale, and other effects that challenge intuition. More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes , which includes quantum corrections to the metric tensor , such as the worldsheet instantons . For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle.
In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase "quantum geometry" usually refers to the formalism within LQG where the observables that capture the information about the geometry are well-defined operators on a Hilbert space . In particular, certain physical observables , such as the area, have a discrete spectrum . LQG is non-commutative . [ 1 ]
It is possible (but considered unlikely) that this strictly quantized understanding of geometry is consistent with the quantum picture of geometry arising from string theory.
Another approach, which tries to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity .
Differential forms are used to express quantum states , using the wedge product : [ 2 ]
where the position vector is
the differential volume element is
and x 1 , x 2 , x 3 are an arbitrary set of coordinates, the upper indices indicate contravariance , lower indices indicate covariance , so explicitly the quantum state in differential form is:
The overlap integral is given by:
in differential form this is
The probability of finding the particle in some region of space R is given by the integral over that region:
provided the wave function is normalized . When R is all of 3d position space, the integral must be 1 if the particle exists.
Differential forms are an approach for describing the geometry of curves and surfaces in a coordinate independent way. In quantum mechanics , idealized situations occur in rectangular Cartesian coordinates , such as the potential well , particle in a box , quantum harmonic oscillator , and more realistic approximations in spherical polar coordinates such as electrons in atoms and molecules . For generality, a formalism which can be used in any coordinate system is useful. | https://en.wikipedia.org/wiki/Quantum_geometry |
In mathematics and physics , a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph ) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) connected at transformer stations (vertices); the differential equations would then describe the voltage along each of the lines, with boundary conditions for each edge provided at the adjacent vertices ensuring that the current added over all edges adds to zero at each vertex.
Quantum graphs were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts, [ 1 ] e.g. as model systems in quantum chaos , in the study of waveguides , in photonic crystals and in Anderson localization , or as a limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology . Another, more simple notion of quantum graphs was introduced by Freedman et al. [ 2 ]
Aside from actually solving the differential equations posed on a quantum graph for purposes of concrete applications, typical questions that arise are those of controllability (what inputs have to be provided to bring the system into a desired state, for example providing sufficient power to all houses on a power network) and identifiability (how and where one has to measure something to obtain a complete picture of the state of the system, for example measuring the pressure of a water pipe network to determine whether or not there is a leaking pipe).
A metric graph is a graph consisting of a set V {\displaystyle V} of vertices and
a set E {\displaystyle E} of edges where each edge e = ( v 1 , v 2 ) ∈ E {\displaystyle e=(v_{1},v_{2})\in E} has been associated
with an interval [ 0 , L e ] {\displaystyle [0,L_{e}]} so that x e {\displaystyle x_{e}} is the coordinate on the
interval, the vertex v 1 {\displaystyle v_{1}} corresponds to x e = 0 {\displaystyle x_{e}=0} and v 2 {\displaystyle v_{2}} to x e = L e {\displaystyle x_{e}=L_{e}} or vice versa. The choice of which vertex lies at zero is
arbitrary with the alternative corresponding to a change of coordinate on the
edge.
The graph has a natural metric: for two
points x , y {\displaystyle x,y} on the graph, ρ ( x , y ) {\displaystyle \rho (x,y)} is
the shortest distance between them
where distance is measured along the edges of the graph.
Open graphs: in the combinatorial graph model
edges always join pairs of vertices however in a quantum graph one may also
consider semi-infinite edges. These are edges associated with the interval [ 0 , ∞ ) {\displaystyle [0,\infty )} attached to a single vertex at x e = 0 {\displaystyle x_{e}=0} .
A graph with one or more
such open edges is referred to as an open graph.
Quantum graphs are metric graphs equipped with a differential
(or pseudo-differential ) operator acting on functions on the graph.
A function f {\displaystyle f} on a metric graph is defined as the | E | {\displaystyle |E|} -tuple of functions f e ( x e ) {\displaystyle f_{e}(x_{e})} on the intervals.
The Hilbert space of the graph is ⨁ e ∈ E L 2 ( [ 0 , L e ] ) {\displaystyle \bigoplus _{e\in E}L^{2}([0,L_{e}])} where the inner product of two functions is
L e {\displaystyle L_{e}} may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the Laplace operator . The operator on an edge is − d 2 d x e 2 {\displaystyle -{\frac {{\textrm {d}}^{2}}{{\textrm {d}}x_{e}^{2}}}} where x e {\displaystyle x_{e}} is the coordinate on the edge. To make the operator self-adjoint a suitable domain must be specified. This is typically achieved by taking the Sobolev space H 2 {\displaystyle H^{2}} of functions on the edges of the graph and specifying matching conditions at the vertices.
The trivial example of matching conditions that make the operator self-adjoint are the Dirichlet boundary conditions , f e ( 0 ) = f e ( L e ) = 0 {\displaystyle f_{e}(0)=f_{e}(L_{e})=0} for every edge. An eigenfunction on a finite edge may be written as
for integer n {\displaystyle n} . If the graph is closed with no infinite edges and the
lengths of the edges of the graph are rationally independent
then an eigenfunction is supported on a single graph edge
and the eigenvalues are n 2 π 2 L e 2 {\displaystyle {\frac {n^{2}\pi ^{2}}{L_{e}^{2}}}} . The Dirichlet conditions
don't allow interaction between the intervals so the spectrum is the same as
that of the set of disconnected edges.
More interesting self-adjoint matching conditions that allow interaction between edges are the Neumann or natural matching conditions. A function f {\displaystyle f} in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero,
where f ′ ( v ) = f ′ ( 0 ) {\displaystyle f'(v)=f'(0)} if the vertex v {\displaystyle v} is at x = 0 {\displaystyle x=0} and f ′ ( v ) = − f ′ ( L e ) {\displaystyle f'(v)=-f'(L_{e})} if v {\displaystyle v} is at x = L e {\displaystyle x=L_{e}} .
The properties of other operators on metric graphs have also been studied.
where A e {\displaystyle A_{e}} is a "magnetic vector potential" on the edge and V e {\displaystyle V_{e}} is a scalar potential.
All self-adjoint matching conditions of the Laplace operator on a graph can be classified according to a scheme of Kostrykin and Schrader. In practice, it is often more convenient to adopt a formalism introduced by Kuchment, see, [ 3 ] which automatically yields an operator in variational form.
Let v {\displaystyle v} be a vertex with d {\displaystyle d} edges emanating from it. For simplicity we choose the coordinates on the edges so that v {\displaystyle v} lies at x e = 0 {\displaystyle x_{e}=0} for each edge meeting at v {\displaystyle v} . For a function f {\displaystyle f} on the graph let
Matching conditions at v {\displaystyle v} can be specified by a pair of matrices A {\displaystyle A} and B {\displaystyle B} through the linear equation,
The matching conditions define a self-adjoint operator if ( A , B ) {\displaystyle (A,B)} has the maximal rank d {\displaystyle d} and A B ∗ = B A ∗ . {\displaystyle AB^{*}=BA^{*}.}
The spectrum of the Laplace operator on a finite graph can be conveniently described
using a scattering matrix approach introduced by Kottos and Smilansky
. [ 4 ] [ 5 ] The eigenvalue problem on an edge is,
So a solution on the edge can be written as a linear combination of plane waves .
where in a time-dependent Schrödinger equation c {\displaystyle c} is the coefficient
of the outgoing plane wave at 0 {\displaystyle 0} and c ^ {\displaystyle {\hat {c}}} coefficient of the incoming
plane wave at 0 {\displaystyle 0} .
The matching conditions at v {\displaystyle v} define a scattering matrix
The scattering matrix relates the vectors of incoming and outgoing plane-wave
coefficients at v {\displaystyle v} , c = S ( k ) c ^ {\displaystyle \mathbf {c} =S(k){\hat {\mathbf {c} }}} .
For self-adjoint matching conditions S {\displaystyle S} is unitary. An element of σ ( u v ) ( v w ) {\displaystyle \sigma _{(uv)(vw)}} of S {\displaystyle S} is a complex transition amplitude
from a directed edge ( u v ) {\displaystyle (uv)} to the edge ( v w ) {\displaystyle (vw)} which in general depends on k {\displaystyle k} .
However, for a large class of matching conditions
the S-matrix is independent of k {\displaystyle k} .
With Neumann matching conditions for example
Substituting in the equation for S {\displaystyle S} produces k {\displaystyle k} -independent transition amplitudes
where δ u w {\displaystyle \delta _{uw}} is the Kronecker delta function that is one if u = w {\displaystyle u=w} and
zero otherwise. From the transition amplitudes we may define a 2 | E | × 2 | E | {\displaystyle 2|E|\times 2|E|} matrix
U {\displaystyle U} is called the bond scattering matrix and
can be thought of as a quantum evolution operator on the graph. It is
unitary and acts on the vector of 2 | E | {\displaystyle 2|E|} plane-wave coefficients for the
graph where c ( u v ) {\displaystyle c_{(uv)}} is the coefficient of
the plane wave traveling from u {\displaystyle u} to v {\displaystyle v} .
The phase e i k L ( u v ) {\displaystyle {\textrm {e}}^{ikL_{(uv)}}} is the phase acquired by the plane wave
when propagating from vertex u {\displaystyle u} to vertex v {\displaystyle v} .
Quantization condition: An eigenfunction on the graph
can be defined through its associated 2 | E | {\displaystyle 2|E|} plane-wave coefficients.
As the eigenfunction is stationary under the quantum evolution a quantization
condition for the graph can be written using the evolution operator.
Eigenvalues k j {\displaystyle k_{j}} occur at values of k {\displaystyle k} where the matrix U ( k ) {\displaystyle U(k)} has an
eigenvalue one. We will order the spectrum with 0 ⩽ k 0 ⩽ k 1 ⩽ … {\displaystyle 0\leqslant k_{0}\leqslant k_{1}\leqslant \dots } .
The first trace formula for a graph was derived by Roth (1983).
In 1997 Kottos and Smilansky used the quantization condition above to obtain
the following trace formula for the Laplace operator on a graph when the
transition amplitudes are independent of k {\displaystyle k} .
The trace formula links the spectrum with periodic orbits on the graph.
d ( k ) {\displaystyle d(k)} is called the density of states. The right hand side of the trace
formula is made up of two terms, the Weyl
term L π {\displaystyle {\frac {L}{\pi }}} is the mean separation of eigenvalues and the oscillating part is a sum
over all periodic orbits p = ( e 1 , e 2 , … , e n ) {\displaystyle p=(e_{1},e_{2},\dots ,e_{n})} on the graph. L p = ∑ e ∈ p L e {\displaystyle L_{p}=\sum _{e\in p}L_{e}} is the length of the orbit and L = ∑ e ∈ E L e {\displaystyle L=\sum _{e\in E}L_{e}} is
the total length of the graph. For an orbit generated by repeating a
shorter primitive orbit, r p {\displaystyle r_{p}} counts the number of repartitions. A p = σ e 1 e 2 σ e 2 e 3 … σ e n e 1 {\displaystyle A_{p}=\sigma _{e_{1}e_{2}}\sigma _{e_{2}e_{3}}\dots \sigma _{e_{n}e_{1}}} is
the product of the transition amplitudes at the vertices of the graph around
the orbit.
Quantum graphs were first employed in the 1930s
to model the spectrum of free electrons in organic molecules like Naphthalene , see figure. As a first approximation the
atoms are taken to be vertices while the
σ-electrons form bonds that fix a frame
in the shape of the molecule on which the free electrons are confined.
A similar problem appears when considering quantum waveguides. These
are mesoscopic systems - systems built with a width on the scale of
nanometers. A quantum waveguide can be thought of as a fattened graph
where the edges
are thin tubes. The spectrum of the Laplace operator on this domain
converges to the spectrum of the Laplace operator on the graph
under certain conditions. Understanding mesoscopic systems plays an
important role in the field of nanotechnology .
In 1997 [ 6 ] Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos , the quantum mechanics of systems that
are classically chaotic. Classical motion on the graph can be defined as
a probabilistic Markov chain where the probability of scattering
from edge e {\displaystyle e} to edge f {\displaystyle f} is given by the absolute value of the
quantum transition amplitude squared, | σ e f | 2 {\displaystyle |\sigma _{ef}|^{2}} . For almost all
finite connected
quantum graphs the probabilistic dynamics is ergodic and mixing,
in other words chaotic.
Quantum graphs embedded in two or three dimensions appear in the study
of photonic crystals . [ 7 ] In two dimensions a simple model of
a photonic crystal consists of polygonal cells of a dense dielectric with
narrow interfaces between the cells filled with air. Studying
dielectric modes that stay mostly in the dielectric gives rise to a
pseudo-differential operator on the graph that follows the narrow interfaces.
Periodic quantum graphs like the lattice in R 2 {\displaystyle {\mathbb {R} }^{2}} are common models of
periodic systems and quantum graphs have been applied
to the study the phenomena of Anderson localization where localized
states occur at the edge of spectral bands in the presence of disorder. | https://en.wikipedia.org/wiki/Quantum_graph |
In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid . In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality ), in its groupoid algebra or in the commutative Hopf algebroid of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous) monoidal categories , Hopf algebroids etc.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_groupoid |
A quantum gyroscope is a very sensitive device to measure angular rotation based on quantum mechanical principles. The first of these was built by Richard Packard and his colleagues at the University of California , Berkeley. The extreme sensitivity means that theoretically, a larger version could detect effects like minute changes in the rotational rate of the Earth.
In 1962, Cambridge University PhD student Brian Josephson hypothesized that an electric current could travel between two superconducting materials even when they were separated by a thin insulating layer.
The term Josephson effect has come to refer generically to the different behaviors that occur in any two weakly connected macroscopic quantum systems—systems composed of molecules that all possess identical wavelike properties.
Among other things, the Josephson effect means that when two superfluids (zero friction fluids) are connected using a weak link and pressure is applied to the superfluid on one side of a weak link, the fluid will oscillate from one side of the weak link to the other. [ citation needed ]
This phenomenon, known as quantum whistling, occurs when pressure is applied to push a superfluid through a very small hole, somewhat as sound is produced by blowing air through an ordinary whistle . A ring-shaped tube full of superfluid, blocked by a barrier containing a tiny hole, could in principle be used to detect pressure differences caused by changes in rotational motion of the ring, in effect functioning as a sensitive gyroscope . Superfluid whistling was first demonstrated using helium-3 , which has the disadvantage of being scarce and expensive, and requiring extremely low temperature (a few thousandths of a Kelvin). Common helium-4 , which remains superfluid at 2 Kelvin, is much more practical, but its quantum whistling is too weak to be heard with a single practical-sized hole. This problem was overcome by using barriers with thousands of holes, in effect a chorus of quantum whistles producing sound waves that reinforced one another by constructive interference . [ citation needed ]
Where Ω {\displaystyle \Omega } is the rotation vector, A is the area vector, and κ s {\displaystyle \kappa _{s}} is the quantum of circulation of helium-3.
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_gyroscope |
Quantum hadrodynamics ( QHD ) [ 1 ] is an effective field theory pertaining to interactions between hadrons , that is, hadron-hadron interactions or the inter-hadron force. It is "a framework for describing the nuclear many-body problem as a relativistic system of baryons and mesons". [ 1 ] Quantum hadrodynamics is closely related and partly derived from quantum chromodynamics , which is the theory of interactions between quarks and gluons that bind them together to form hadrons, via the strong force .
An important phenomenon in quantum hadrodynamics is the nuclear force , or residual strong force. It is the force operating between those hadrons which are nucleons – protons and neutrons – as it binds them together to form the atomic nucleus . The bosons which mediate the nuclear force are three types of mesons : pions , rho mesons and omega mesons . Since mesons are themselves hadrons, quantum hadrodynamics also deals with the interaction between the carriers of the nuclear force itself, alongside the nucleons bound by it. The hadrodynamic force keeps nuclei bound, against the electrodynamic force which operates to break them apart (due to the mutual repulsion between protons in the nucleus).
Quantum hadrodynamics, dealing with the nuclear force and its mediating mesons, can be compared to other quantum field theories which describe fundamental forces and their associated bosons: quantum chromodynamics, dealing with the strong interaction and gluons; quantum electrodynamics , dealing with electromagnetism and photons ; quantum flavordynamics , dealing with the weak interaction and W and Z bosons .
This particle physics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_hadrodynamics |
A quantum heat engine is a device that generates power from the heat flow between hot and cold reservoirs.
The operation mechanism of the engine can be described by the laws of quantum mechanics .
The first realization of a quantum heat engine was pointed out by Scovil and Schulz-DuBois in 1959, [ 1 ] showing the connection of efficiency of the Carnot engine and the 3-level maser .
Quantum refrigerators share the structure of quantum heat engines with the purpose of pumping heat from a cold to a hot bath consuming power
first suggested by Geusic, Schulz-DuBois, De Grasse and Scovil. [ 2 ] When the power is supplied by a laser the process is termed optical pumping or laser cooling , suggested by Wineland and Hänsch . [ 3 ] [ 4 ] [ 5 ] Surprisingly heat engines and refrigerators can operate up to the scale of a single particle thus justifying the need for a quantum theory termed quantum thermodynamics . [ 6 ]
The three-level-amplifier is the template of a quantum device. It operates by employing a hot and cold bath
to maintain population inversion between two energy levels which is used to amplify light by stimulated emission [ 7 ] The ground state level ( 1-g ) and the excited level ( 3-h ) are coupled to a hot bath of temperature T h {\displaystyle T_{\text{h}}} .
The energy gap is ℏ ω h = E 3 − E 1 {\displaystyle \hbar \omega _{\text{h}}=E_{3}-E_{1}} . When the population on the levels equilibrate
where ℏ = h 2 π {\displaystyle \hbar ={\frac {h}{2\pi }}} is the Planck constant and k B {\displaystyle k_{\text{B}}} is the Boltzmann constant .
The cold bath of temperature T c {\displaystyle T_{\text{c}}} couples the ground ( 1-g ) to an intermediate level ( 2-c )
with energy gap E 2 − E 1 = ℏ ω c {\displaystyle E_{2}-E_{1}=\hbar \omega _{\text{c}}} .
When levels 2-c and 1-g equilibrate then
The device operates as an amplifier when levels ( 3-h ) and ( 2-c ) are coupled to an external field of frequency ν {\displaystyle \nu } .
For optimal resonance conditions ν = ω h − ω c {\displaystyle \nu =\omega _{\text{h}}-\omega _{\text{c}}} .
The efficiency of the amplifier in converting heat to power is the ratio of work output to heat input:
Amplification of the field is possible only for positive gain (population inversion) G = N h − N c ≥ 0 {\displaystyle G=N_{\text{h}}-N_{\text{c}}\geq 0} .
This is equivalent to ℏ ω c k B T c ≥ ℏ ω h k B T h {\displaystyle {\frac {\hbar \omega _{\text{c}}}{k_{\text{B}}T_{\text{c}}}}\geq {\frac {\hbar \omega _{\text{h}}}{k_{\text{B}}T_{\text{h}}}}} .
Inserting this expression into the efficiency formula leads to:
where η c {\displaystyle \eta _{\text{c}}} is the Carnot cycle efficiency .
Equality is obtained under a zero gain condition G = 0 {\displaystyle G=0} .
The relation between the quantum amplifier and the Carnot efficiency was first pointed out by Scovil and Schultz-DuBois: [ 1 ]
Reversing the operation driving heat from the cold bath to the hot bath by consuming power constitutes a refrigerator .
The efficiency of the refrigerator defined as the coefficient of performance (COP) for the reversed device is:
Quantum devices can operate either continuously or by a reciprocating cycle.
Continuous devices include solar cells converting solar radiation to electrical power, thermoelectric where the output is current and lasers where the output power is coherent light.
The primary example of a continuous refrigerator is optical pumping and laser cooling . [ 8 ] [ 9 ] Similarly to classical reciprocating engines, quantum heat engines also have a cycle that is divided into different strokes. A stroke is time segment in which a certain operation takes place (e.g. thermalization, or work extraction). Two adjacent strokes do not commute with each other. The most common reciprocating heat machines are the four-stroke machine, and the two-stroke machine. Reciprocating devices have been suggested operating either by the Carnot cycle [ 10 ] [ 11 ] or the Otto cycle . [ 12 ]
In both types the quantum description allows to obtain equation of motion for the working medium
and the heat flow from the reservoirs.
Quantum versions of most of the common thermodynamic cycles have been studied, for example the Carnot cycle , [ 10 ] [ 11 ] [ 13 ] Stirling cycle [ 14 ] and Otto cycle . [ 12 ] [ 15 ]
The Otto cycle can serve as a template for other reciprocating cycles.
It is composed of the following four segments:
The propagator of the four stroke cycle
becomes U global {\displaystyle U_{\text{global}}} , which is the ordered product of the segment propagators:
The propagators are linear operators defined on a vector space which completely determines the state of the working medium.
Common to all thermodynamic cycles the consecutive segment propagators do not commute [ U i , U j ] ≠ 0 {\displaystyle [{\ U}_{i},{U}_{j}]\neq 0} .
Commuting propagators will lead to zero power.
In a reciprocating quantum heat engine the working medium is a quantum system such as spin systems [ 16 ] or an harmonic oscillator. [ 17 ] For maximum power the cycle time should be optimized.
There are two basic timescales in the reciprocating refrigerator the cycle time τ cyc {\displaystyle \tau _{\text{cyc}}} and the internal
timescale 2 π / ω {\displaystyle 2\pi /\omega } . In general when τ cyc ≫ 2 π / ω {\displaystyle \tau _{\text{cyc}}\gg 2\pi /\omega } the
engine operates in quasi-adiabatic conditions. The only quantum effect can be found at low temperatures
where the unit of energy of the device becomes ℏ ω {\displaystyle \hbar \omega } instead of k B T {\displaystyle k_{\text{B}}T} .
The efficiency at this limit is η = 1 − ω c ω h {\displaystyle \eta =1-{\frac {\omega _{\text{c}}}{\omega _{\text{h}}}}} , always smaller than the Carnot efficiency η c {\displaystyle \eta _{\text{c}}} . At high temperature and for the harmonic working medium the efficiency at maximum power becomes η = 1 − T c T h {\displaystyle \eta =1-{\sqrt {\frac {T_{\text{c}}}{T_{\text{h}}}}}} which is the endoreversible thermodynamics result. [ 17 ]
For shorter cycle times the working medium cannot follow adiabatically the change in the external parameter.
This leads to friction-like phenomena. Extra power is required to drive the system faster.
The signature of such dynamics is the development of coherence causing extra dissipation.
Surprisingly the dynamics leading to friction is quantized meaning that frictionless solutions to the adiabatic expansion /compression
can be found in finite time. [ 18 ] [ 19 ] As a result, optimization has to be carried out only with respect to the time allocated
to heat transport. In this regime the quantum feature of coherence degrades the performance.
Optimal frictionless performance is obtained when the coherence can be cancelled.
The shortest cycle times τ cyc ≪ 2 π / ω {\displaystyle \tau _{\text{cyc}}\ll 2\pi /\omega } , sometimes termed sudden cycles, [ 20 ] have universal features. In this case coherence contributes to the cycles power.
A two-stroke engine quantum cycle equivalent to the Otto cycle based on two qubits has been proposed.
The first qubit has frequency ω h {\displaystyle \omega _{\text{h}}} and the second ω c {\displaystyle \omega _{\text{c}}} . The cycle is composed
of a first stroke of partial equilibration of the two qubits with the hot and cold bath in parallel.
The second power stroke is composed of a partial or full swap between the qubits.
The swap operation is generated by a unitary transformation which preserves the entropy as a result it is a pure power stroke. [ 21 ] [ 22 ]
The quantum Otto cycle refrigerators shares the same cycle with magnetic refrigeration . [ 23 ]
Continuous quantum engines are the quantum analogues of turbines . The work output mechanism is coupling to an external periodic field, typically the electromagnetic field. Thus the heat engine is a model for a laser . [ 9 ] The models differ by the choice of their working substance
and heat source and sink. Externally driven two-level, [ 24 ] three level [ 25 ] four-level [ 26 ] [ 27 ] and coupled harmonic oscillators [ 28 ] have been studied.
The periodic driving splits the energy level structure of the working medium. This splitting allows the two level engine to couple
selectively to the hot and cold baths and produce power. On the other hand, ignoring this splitting in the derivation of the equation of motion will violate the second law of thermodynamics . [ 29 ]
Non thermal fuels have been considered for quantum heat engines. The idea is to increase the energy content of the hot bath without
increasing its entropy. This can be achieved by employing coherence [ 30 ] or a squeezed thermal bath. [ 31 ] These devices do not violate the second law of thermodynamics.
Two-stroke, Four-stroke, and continuous machine are very different from each other. However it was shown [ 32 ] that there is a quantum regime where all these machines become thermodynamically equivalent to each other. While the intra cycle dynamics in the equivalence regime is very different in different engine types, when the cycle is completed they all turn out to provide the same amount of work and consume the same amount of heat (hence they share the same efficiency as well). This equivalence is associated with a coherent work extraction mechanism and has no classical analogue. These quantum features have been demonstrated experimentally. [ 33 ]
The elementary example operates under quasi equilibrium conditions. Its main quantum feature is the discrete energy level structure.
More realistic devices operate out of equilibrium possessing friction heat leaks and finite heat flow. Quantum thermodynamics supplies a dynamical theory required for systems out of equilibrium such as heat engines, thus,
inserting dynamics into thermodynamics.
The theory of open quantum systems constitutes the basic theory. For heat engines a reduced description of the dynamics
of the working substance is sought, tracing out the hot and cold baths.
The starting point is the general Hamiltonian of the combined systems:
and the system Hamiltonian H s ( t ) {\displaystyle H_{\text{s}}(t)} is time dependent.
A reduced description leads to the equation of motion of the system:
where ρ {\displaystyle \rho } is the density operator describing the state of the working medium and L h/c {\displaystyle L_{\text{h/c}}} is the generator of dissipative dynamics
which includes the heat transport terms from the baths.
Using this construction, the total change in energy of the sub-system becomes:
leading to the dynamical version of the first law of thermodynamics : [ 6 ]
The rate of entropy production becomes:
The global structure of quantum mechanics is reflected in the derivation of the reduced description.
A derivation which is consistent with the laws of thermodynamics is based on
the weak coupling limit.
A thermodynamical idealization assumes that the system and the baths are uncorrelated, meaning that the total state
of the combined system becomes a tensor product at all times:
Under these conditions the dynamical equations of motion become: d d t ρ s = L ρ s , {\displaystyle {\frac {d}{dt}}\rho _{\text{s}}={L}\rho _{\text{s}}~,} where L {\displaystyle {L}} is the Liouville superoperator described in terms of the system's Hilbert space,
where the reservoirs are described implicitly.
Within the formalism of quantum open system, L {\displaystyle L} can take the form of the
Gorini-Kossakowski-Sudarshan-Lindblad (GKS-L) Markovian generator or also known just as Lindblad equation . [ 34 ] Theories beyond the weak coupling regime have been proposed. [ 35 ] [ 36 ] [ 37 ]
The absorption refrigerator is of unique importance in setting an autonomous quantum device.
Such a device requires no external power and operates without external intervention in scheduling the operations
. [ 38 ] [ 39 ] [ 40 ] The basic construct includes three baths; a power bath, a hot bath and a cold bath.
The tricycle model is the template for the absorption refrigerator.
The tricycle engine has a generic structure.
The basic model consists of three thermal baths: A hot bath with temperature T h {\displaystyle T_{\text{h}}} ,
a cold bath with temperature T c {\displaystyle T_{\text{c}}} and a work bath with temperature T d {\displaystyle T_{\text{d}}} .
Each bath is connected to the engine via a frequency filter which can be modeled by three oscillators:
where ω h {\displaystyle \omega _{\text{h}}} , ω c {\displaystyle \omega _{\text{c}}} and ω d {\displaystyle \omega _{\text{d}}} are the filter frequencies on resonance ω d = ω h − ω c {\displaystyle \omega _{\text{d}}=\omega _{\text{h}}-\omega _{\text{c}}} .
The device operates as a refrigerator by removing an excitation from the cold bath as well as from the work bath
and generating an excitation in the hot bath. The term a † b c {\displaystyle a^{\dagger }bc} in the Hamiltonian is non linear
and crucial for an engine or a refrigerator.
where ϵ {\displaystyle \epsilon } is the coupling strength.
The first-law of thermodynamics represents the energy balance of heat currents originating from the three baths and collimating on the system:
At steady state no heat is accumulated in the tricycle, thus d E s d t = 0 {\displaystyle {\frac {dE_{\text{s}}}{dt}}=0} .
In addition, in steady state the entropy is only generated in the baths, leading to the second law of thermodynamics :
This version of the second-law is a generalisation of the statement of Clausius theorem ;
heat does not flow spontaneously from cold to hot bodies.
When the temperature T d → ∞ {\displaystyle T_{\text{d}}\rightarrow \infty } , no entropy is generated in the power bath.
An energy current with no accompanying entropy production is equivalent to generating pure power: P = J d {\displaystyle {P}={J}_{\text{d}}} , where P {\displaystyle {P}} is the power output.
There are seemingly two independent formulations of the third law of thermodynamics both originally were stated by Walther Nernst . The first formulation is known as the Nernst heat theorem , and can be phrased as:
The second formulation is dynamical, known as the unattainability principle : [ 41 ]
At steady state the second law of thermodynamics implies that the total entropy production is non-negative.
When the cold bath approaches the absolute zero temperature,
it is necessary to eliminate the entropy production divergence at the cold side
when T c → 0 {\displaystyle T_{\text{c}}\rightarrow 0} , therefore
For α = 0 {\displaystyle \alpha =0} the fulfillment of the second law depends on the entropy production of the other baths,
which should compensate for the negative entropy production of the cold bath.
The first formulation of the third law modifies this restriction.
Instead of α ≥ 0 {\displaystyle \alpha \geq 0} the third law imposes α > 0 {\displaystyle \alpha >0} ,
guaranteeing that at absolute zero the entropy production at the cold bath is zero: S ˙ c = 0 {\displaystyle {\dot {S}}_{\text{c}}=0} .
This requirement leads to the scaling condition of the heat current J c ∝ T c α + 1 {\displaystyle {J}_{\text{c}}\propto T_{\text{c}}^{\alpha +1}} .
The second formulation, known as the unattainability principle can be rephrased as; [ 42 ]
The dynamics of the cooling process is governed by the equation
where c V ( T c ) {\displaystyle c_{V}(T_{\text{c}})} is the heat capacity of the bath. Taking J c ∝ T c α + 1 {\displaystyle {J}_{\text{c}}\propto T_{\text{c}}^{\alpha +1}} and c V ∼ T c η {\displaystyle c_{V}\sim T_{\text{c}}^{\eta }} with η ≥ 0 {\displaystyle {\eta }\geq 0} , we can quantify this formulation by evaluating the characteristic exponent ζ {\displaystyle \zeta } of the cooling process,
This equation introduce the relation between the characteristic exponents ζ {\displaystyle \zeta } and α {\displaystyle \alpha } . When ζ < 0 {\displaystyle \zeta <0} then the bath is cooled to zero temperature in a finite time, which implies a violation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem .
Deffner, Sebastian and Campbell, Steve. "Quantum Thermodynamics: An introduction to the thermodynamics of quantum information", (Morgan & Claypool Publishers, 2019). [ 1 ]
F. Binder, L. A. Correa, C. Gogolin, J. Anders, G. Adesso (eds.) "Thermodynamics in the Quantum Regime. Fundamental Aspects and New Directions." (Springer 2018)
Gemmer, Jochen, M. Michel, and Günter Mahler. "Quantum thermodynamics. Emergence of thermodynamic behavior within composite quantum systems. 2." (2009).
Petruccione, Francesco, and Heinz-Peter Breuer. The theory of open quantum systems. Oxford university press, 2002. | https://en.wikipedia.org/wiki/Quantum_heat_engines_and_refrigerators |
In condensed matter physics , quantum hydrodynamics ( QHD ) [ 1 ] is most generally the study of hydrodynamic-like systems which demonstrate quantum mechanical behavior. They arise in semiclassical mechanics in the study of metal and semiconductor devices, in which case being derived from the Boltzmann transport equation combined with Wigner quasiprobability distribution . In quantum chemistry they arise as solutions to chemical kinetic systems, in which case they are derived from the Schrödinger equation by way of Madelung equations .
An important system of study in quantum hydrodynamics is that of superfluidity . Some other topics of interest in quantum hydrodynamics are quantum turbulence , quantized vortices , second and third sound , and quantum solvents . The quantum hydrodynamic equation is an equation in Bohmian mechanics , which, it turns out, has a mathematical relationship to classical fluid dynamics (see Madelung equations ).
Some common experimental applications of these studies are in liquid helium ( 3 He and 4 He ), and of the interior of neutron stars and the quark–gluon plasma . Many famous scientists have worked in quantum hydrodynamics, including Richard Feynman , Lev Landau , and Pyotr Kapitsa . | https://en.wikipedia.org/wiki/Quantum_hydrodynamics |
Quantum computation, which exploits quantum parallelism, is in principle faster than a classical computer for certain problems. [ 1 ] Quantum image is encoding the image information in quantum-mechanical systems instead of classical ones and replacing classical with quantum information processing may alleviate some of these challenges. [ 2 ]
Humans obtain most of their information through their eyes. Accordingly, the analysis of visual data is one of the most important functions of our brain and it has evolved high efficiency in processing visual data. Currently, visual information like images and videos constitutes the largest part of data traffic in the internet. Processing of this information requires ever-larger computational power. [ 3 ]
The laws of quantum mechanics allow one to reduce the required resources for some tasks by many orders of magnitude if the image data are encoded in the quantum state of a suitable physical system. [ 4 ] The researchers discuss a suitable method for encoding image data, and develop a new quantum algorithm that can detect boundaries among parts of an image with a single logical operation. This edge-detection operation is independent of the size of the image. Several other algorithms are also discussed. It is theoretically and experimentally demonstrated that they work in practice. This is the first experiment to demonstrate practical quantum image processing. It contributes a substantial progress towards both theoretical and experimental quantum computing for image processing, it will stimulate future studies in the field of quantum information processing of visual data. | https://en.wikipedia.org/wiki/Quantum_image |
Quantum image processing (QIMP) is using quantum computing or quantum information processing to create and work with quantum images . [ 1 ] [ 2 ]
Due to some of the properties inherent to quantum computation, notably entanglement and parallelism , it is hoped that QIMP technologies will offer capabilities and performances that surpass their traditional equivalents, in terms of computing speed, security, and minimum storage requirements. [ 2 ] [ 3 ]
A. Y. Vlasov's work [ 4 ] in 1997 focused on using a quantum system to recognize orthogonal images. This was followed by efforts using quantum algorithms to search specific patterns in binary images [ 5 ] and detect the posture of certain targets. [ 6 ] Notably, more optics-based interpretations for quantum imaging were initially experimentally demonstrated in [ 7 ] and formalized in [ 8 ] after seven years.
In 2003, Salvador Venegas-Andraca and S. Bose presented Qubit Lattice, the first published general model for storing, processing and retrieving images using quantum systems. [ 9 ] [ 10 ] Later on, in 2005, Latorre proposed another kind of representation, called the Real Ket, [ 11 ] whose purpose was to encode quantum images as a basis for further applications in QIMP. Furthermore, in 2010 Venegas-Andraca and Ball presented a method for storing and retrieving binary geometrical shapes in quantum mechanical systems in which it is shown that maximally entangled qubits can be used to reconstruct images without using any additional information. [ 12 ]
Technically, these pioneering efforts with the subsequent studies related to them can be classified into three main groups: [ 3 ]
A survey of quantum image representation has been published in. [ 14 ] Furthermore, the recently published book Quantum Image Processing [ 15 ] provides a comprehensive introduction to quantum image processing, which focuses on extending conventional image processing tasks to the quantum computing frameworks. It summarizes the available quantum image representations and their operations, reviews the possible quantum image applications and their implementation, and discusses the open questions and future development trends.
There are various approaches for quantum image representation, that are usually based on the encoding of color information. A common representation is FRQI ( Flexible Representation for Quantum Images ), that captures the color and position at every pixel of the image, and defined as: [ 16 ] | I ⟩ = 1 2 n ∑ i = 0 2 2 n − 1 | c i ⟩ ⊗ | i ⟩ {\displaystyle \vert I\rangle ={\frac {1}{2^{n}}}\sum _{i=0}^{2^{2n-1}}\vert c_{i}\rangle \otimes \vert i\rangle } where | i ⟩ {\textstyle |i\rangle } is the position and | c i ⟩ = c o s θ i | 0 ⟩ + s i n θ i | 1 ⟩ {\textstyle \vert c_{i}\rangle =cos\theta _{i}\vert 0\rangle +sin\theta _{i}\vert 1\rangle } the color with a vector of angles θ i ∈ [ 0 , π / 2 ] {\textstyle \theta _{i}\in \left[0,\pi /2\right]} . As it can be seen, | c i ⟩ {\textstyle \vert c_{i}\rangle } is a regular qubit state of the form | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ {\displaystyle \vert \psi \rangle =\alpha \vert 0\rangle +\beta \vert 1\rangle } , with basis states | 0 ⟩ = ( 1 0 ) {\textstyle \vert 0\rangle ={\begin{pmatrix}1\\0\end{pmatrix}}} and | 1 ⟩ = ( 0 1 ) {\textstyle \vert 1\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}} , as well as amplitudes α {\textstyle \alpha } and β {\textstyle \beta } that satisfy | α | 2 + | β | 2 = 1 {\textstyle \left|\alpha \right|^{2}+\left|\beta \right|^{2}=1} . [ 17 ]
Another common representation is MCQI ( Multi-Channel Representation for Quantum Images ), that uses the RGB channels with quantum states and following FRQI definition: [ 16 ] | I ⟩ = 1 2 n + 1 ∑ i = 0 2 2 n − 1 | C R G B i ⟩ ⊗ | i ⟩ {\displaystyle \vert I\rangle ={\frac {1}{2^{n+1}}}\sum _{i=0}^{2^{2n-1}}\vert C_{RGB}^{i}\rangle \otimes \vert i\rangle } | C R G B i ⟩ = cos θ R i | 000 ⟩ + cos θ G i | 001 ⟩ + cos θ B i | 010 ⟩ + sin θ R i | 100 ⟩ + sin θ G i | 101 ⟩ + sin θ B i | 110 ⟩ + cos θ α | 011 ⟩ + sin θ α | 111 ⟩ {\displaystyle {\begin{aligned}{\begin{aligned}\vert C_{RGB}^{i}\rangle &={\cos \theta _{R}^{i}\vert 000\rangle }+{\cos \theta _{G}^{i}\vert 001\rangle }+{\cos \theta _{B}^{i}\vert 010\rangle }\\&\quad +{\sin \theta _{R}^{i}\vert 100\rangle }+{\sin \theta _{G}^{i}\vert 101\rangle }+{\sin \theta _{B}^{i}\vert 110\rangle }\\&\quad +{\cos {\theta _{\alpha }}\vert 011\rangle }+{\sin \theta _{\alpha }\vert 111\rangle }\end{aligned}}\end{aligned}}}
Departing from the angle-based approach of FRQI and MCQI, and using a qubit sequence, NEQR ( Novel Enhanced Representation for Quantum Images ) is another representation approach, that uses a function f ( y , x ) = C y x q − 1 C y x q − 2 … C y x 1 C y x 0 {\textstyle f\left(y,x\right)=C_{yx}^{q-1}C_{yx}^{q-2}\ldots C_{yx}^{1}C_{yx}^{0}} to encode color values for a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} image: [ 16 ] | I ⟩ = 1 2 n ∑ y = 0 2 n − 1 ∑ x = 0 2 n − 1 | f ( y , x ) ⟩ | y x ⟩ {\displaystyle \vert I\rangle ={\frac {1}{2^{n}}}\sum _{y=0}^{2^{n}-1}\sum _{x=0}^{2^{n}-1}\vert f\left(y,x\right)\rangle \vert yx\rangle }
A lot of the effort in QIMP has been focused on designing algorithms to manipulate the position and color information encoded using flexible representation of quantum images (FRQI) and its many variants. For instance, FRQI-based fast geometric transformations including (two-point) swapping, flip, (orthogonal) rotations [ 18 ] and restricted geometric transformations to constrain these operations to a specified area of an image [ 19 ] were initially proposed. Recently, NEQR-based quantum image translation to map the position of each picture element in an input image into a new position in an output image [ 20 ] and quantum image scaling to resize a quantum image [ 21 ] were discussed. While FRQI-based general form of color transformations were first proposed by means of the single qubit gates such as X, Z, and H gates. [ 22 ] Later, Multi-Channel Quantum Image-based channel of interest (CoI) operator to entail shifting the grayscale value of the preselected color channel and the channel swapping (CS) operator to swap the grayscale values between two channels have been fully discussed. [ 23 ]
To illustrate the feasibility and capability of QIMP algorithms and application, researchers always prefer to simulate the digital image processing tasks on the basis of the QIRs that we already have. By using the basic quantum gates and the aforementioned operations, so far, researchers have contributed to quantum image feature extraction , [ 24 ] quantum image segmentation , [ 25 ] quantum image morphology, [ 26 ] quantum image comparison, [ 27 ] quantum image filtering, [ 28 ] quantum image classification, [ 29 ] quantum image stabilization , [ 30 ] among others. In particular, QIMP-based security technologies have attracted extensive interest of researchers as presented in the ensuing discussions. Similarly, these advancements have led to many applications in the areas of watermarking , [ 31 ] [ 32 ] [ 33 ] encryption, [ 34 ] and steganography [ 35 ] etc., which form the core security technologies highlighted in this area.
In general, the work pursued by the researchers in this area are focused on expanding the applicability of QIMP to realize more classical-like digital image processing algorithms; propose technologies to physically realize the QIMP hardware; or simply to note the likely challenges that could impede the realization of some QIMP protocols.
By encoding and processing the image information in quantum-mechanical systems, a framework of quantum image processing is presented, where a pure quantum state encodes the image information: to encode the pixel values in the probability amplitudes and the pixel positions in the computational basis states.
Given an image F = ( F i , j ) M × L {\displaystyle F=(F_{i,j})_{M\times L}} , where F i , j {\displaystyle F_{i,j}} represents the pixel value at position ( i , j ) {\displaystyle (i,j)} with i = 1 , … , M {\displaystyle i=1,\dots ,M} and j = 1 , … , L {\displaystyle j=1,\dots ,L} , a vector f → {\displaystyle {\vec {f}}} with M L {\displaystyle ML} elements can be formed by letting the first M {\displaystyle M} elements of f → {\displaystyle {\vec {f}}} be the first column of F {\displaystyle F} , the next M {\displaystyle M} elements the second column, etc.
A large class of image operations is linear , e.g., unitary transformations, convolutions, and linear filtering. In the quantum computing, the linear transformation can be represented as | g ⟩ = U ^ | f ⟩ {\displaystyle |g\rangle ={\hat {U}}|f\rangle } with the input image state | f ⟩ {\displaystyle |f\rangle } and the output image state | g ⟩ {\displaystyle |g\rangle } . A unitary transformation can be implemented as a unitary evolution.
Some basic and commonly used image transforms (e.g., the Fourier , Hadamard , and Haar wavelet transforms) can be expressed in the form G = P F Q {\displaystyle G=PFQ} , with the resulting image G {\displaystyle G} and a row (column) transform matrix P ( Q ) {\displaystyle P(Q)} .
The corresponding unitary operator U ^ {\displaystyle {\hat {U}}} can then be written as U ^ = Q T ⊗ P {\displaystyle {\hat {U}}={Q}^{T}\otimes {P}} . Several commonly used two-dimensional image transforms, such as the Haar wavelet, Fourier, and Hadamard transforms, are experimentally demonstrated on a quantum computer, [ 36 ] with exponential speedup over their classical counterparts. In addition, a novel highly efficient quantum algorithm is proposed and experimentally implemented for detecting the boundary between different regions of a picture: It requires only one single-qubit gate in the processing stage, independent of the size of the picture. | https://en.wikipedia.org/wiki/Quantum_image_processing |
Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system , that has become one of the characteristics of the standard description of quantum physics . Prior to quantum physics, it was thought that
Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes of measurements of an observable . The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.
Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. By the later half of the 18th century, measurement errors were well understood, and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.
An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of quantum mechanics and quantum measurement continues to be an active research area in both theoretical and experimental physics. [ 1 ] Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann . The kinds of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of projection-valued measures for self-adjoint operators that had been recently developed (by von Neumann and independently by Marshall Stone ) and the Hilbert space formulation of quantum mechanics (attributed by von Neumann to Paul Dirac ).
In this formulation, the state of a physical system corresponds to a vector of length 1 in a Hilbert space H over the complex numbers . An observable is represented by a self-adjoint (i.e. Hermitian ) operator A on H . If H is finite dimensional , by the spectral theorem , A has an orthonormal basis of eigenvectors . If the system is in state ψ , then immediately after measurement the system will occupy a state that is an eigenvector e of A and the observed value λ will be the corresponding eigenvalue of the equation Ae = λe . It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ψ . The probability is Pr ( λ ) = ⟨ E ( λ ) ψ ∣ ψ ⟩ {\displaystyle \operatorname {Pr} (\lambda )=\langle \operatorname {E} (\lambda )\psi \mid \psi \rangle } where E ( λ ) is the projection onto the space of eigenvectors of A with eigenvalue λ .
In this example, we consider a single spin 1/2 particle (such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space C 2 , with each quantum state corresponding to a unit vector in C 2 (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.
The Pauli spin matrices σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 − i i 0 ) , σ 3 = ( 1 0 0 − 1 ) {\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}} are self-adjoint and correspond to spin-measurements along the 3 coordinate axes.
The Pauli matrices all have the eigenvalues +1, −1.
Thus in the state ψ = 1 2 ( 1 , 1 ) , {\displaystyle \psi ={\frac {1}{\sqrt {2}}}(1,1),} σ 1 has the determinate value +1, while measurement of σ 3 can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both σ 1 and σ 3 have determinate values.
There are various questions that can be asked about the above indeterminacy assertion.
Von Neumann formulated the question 1) and provided an argument why the answer had to be no, if one accepted the formalism he was proposing. However, according to Bell, von Neumann's formal proof did not justify his informal conclusion. [ 2 ] A definitive but partial negative answer to 1) has been established by experiment: because Bell's inequalities are violated, any such hidden variable(s) cannot be local (see Bell test experiments ).
The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the observer effect , which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) that measures exclusively σ 1 and (B) that measures only σ 3 of a spin system in the state ψ . The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with equal probability.
Quantum indeterminacy can also be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified. This quantum uncertainty principle can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy.
The magnitude involved in quantum uncertainty is on the order of the Planck constant ( 6.626 070 15 × 10 −34 J⋅Hz −1 [ 3 ] ).
Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed, according to the Kochen–Specker theorem , in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable properties ( observables ) has a determinate (sharp) value. The values of an observable will be obtained non-deterministically in accordance with a probability distribution that is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.
This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ 1 as a filter that retains only those particles such that σ 1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ .
However, Albert Einstein believed that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein, Boris Podolsky and Nathan Rosen showed that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas:
This failure of the classical view was one of the conclusions of the EPR thought experiment in which two remotely located observers , now commonly referred to as Alice and Bob , perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was a conclusion of EPR, using the formal apparatus of quantum theory, that once Alice measured spin in the x direction, Bob's measurement in the x direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. From this it follows that either value of spin in the x direction is not an element of reality or that the effect of Alice's measurement has infinite speed of propagation.
We have described indeterminacy for a quantum system that is in a pure state . Mixed states are a more general kind of state obtained by a statistical mixture of pure states. For mixed states
the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:
Let A be an observable of a quantum mechanical system. A is given by a densely
defined self-adjoint operator on H . The spectral measure of A is a projection-valued measure defined by the condition
for every Borel subset U of R . Given a mixed state S , we introduce the distribution of A under S as follows:
This is a probability measure defined on the Borel subsets of R that is the probability distribution obtained by measuring A in S .
Quantum indeterminacy is often understood as information (or lack of it) whose existence we infer, occurring in individual quantum systems, prior to measurement. Quantum randomness is the statistical manifestation of that indeterminacy, witnessable in results of experiments repeated many times. However, the relationship between quantum indeterminacy and randomness is subtle and can be considered differently. [ 4 ]
In classical physics , experiments of chance, such as coin-tossing and dice-throwing, are deterministic, in the sense that, perfect knowledge of the initial conditions would render outcomes perfectly predictable. The ‘randomness’ stems from ignorance of physical information in the initial toss or throw. In diametrical contrast, in the case of quantum physics , the theorems of Kochen and Specker, [ 5 ] the inequalities of John Bell, [ 6 ] and experimental evidence of Alain Aspect , [ 7 ] [ 8 ] all indicate that quantum randomness does not stem from any such physical information .
In 2008, Tomasz Paterek et al. provided an explanation in mathematical information . They proved that quantum randomness is, exclusively, the output of measurement experiments whose input settings introduce logical independence into quantum systems. [ 9 ] [ 10 ]
Logical independence is a well-known phenomenon in Mathematical Logic . It refers to the null logical connectivity that exists between mathematical propositions (in the same language) that neither prove nor disprove one another. [ 11 ]
In the work of Paterek et al., the researchers demonstrate a link connecting quantum randomness and logical independence in a formal system of Boolean propositions. In experiments measuring photon polarisation, Paterek et al. demonstrate statistics correlating predictable outcomes with logically dependent mathematical propositions, and random outcomes with propositions that are logically independent. [ 12 ] [ 13 ]
In 2020, Steve Faulkner reported on work following up on the findings of Tomasz Paterek et al.; showing what logical independence in the Paterek Boolean propositions means, in the domain of Matrix Mechanics proper. He showed how indeterminacy's indefiniteness arises in evolved density operators representing mixed states, where measurement processes encounter irreversible 'lost history' and ingression of ambiguity. [ 14 ] | https://en.wikipedia.org/wiki/Quantum_indeterminacy |
Quantum information is the information of the state of a quantum system . It is the basic entity of study in quantum information theory , [ 1 ] [ 2 ] [ 3 ] and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.
It is an interdisciplinary field that involves quantum mechanics , computer science , information theory , philosophy and cryptography among other fields. [ 4 ] [ 5 ] [ 6 ] Its study is also relevant to disciplines such as cognitive science , psychology and neuroscience . [ 7 ] [ 8 ] [ 9 ] [ 10 ] Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method . In quantum mechanics , due to the uncertainty principle , non-commuting observables cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of the system is an eigenstate of the observable. [ 11 ] Since any two non-commuting observables are not simultaneously well-defined, a quantum state can never contain definitive information about both non-commuting observables. [ 8 ]
Data can be encoded into the quantum state of a quantum system as quantum information . [ 12 ] While quantum mechanics deals with examining properties of matter at the microscopic level, [ 13 ] [ 8 ] quantum information science focuses on extracting information from those properties, [ 8 ] and quantum computation manipulates and processes information – performs logical operations – using quantum information processing techniques. [ 14 ]
Quantum information, like classical information, can be processed using digital computers , transmitted from one location to another, manipulated with algorithms , and analyzed with computer science and mathematics . Just like the basic unit of classical information is the bit, quantum information deals with qubits . [ 15 ] Quantum information can be measured using Von Neumann entropy.
Recently, the field of quantum computing has become an active research area because of the possibility to disrupt modern computation, communication, and cryptography . [ 14 ] [ 16 ]
The history of quantum information theory began at the turn of the 20th century when classical physics was revolutionized into quantum physics . The theories of classical physics were predicting absurdities such as the ultraviolet catastrophe , or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypotheses to classical physics. Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics was born. [ 2 ]
Quantum mechanics was formulated by Erwin Schrödinger using wave mechanics and Werner Heisenberg using matrix mechanics . [ 17 ] The equivalence of these methods was proven later. [ 18 ] Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von Neumann formulated quantum theory using operator algebra in a way that it described measurement as well as dynamics. [ 19 ] These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.
See: Dynamical Pictures
In the 1960s, Ruslan Stratonovich , Carl Helstrom and Gordon [ 20 ] proposed a formulation of optical communications using quantum mechanics. This was the first historical appearance of quantum information theory. They mainly studied error probabilities and channel capacities for communication. [ 20 ] [ 21 ] [ 22 ] Later, Alexander Holevo obtained an upper bound of communication speed in the transmission of a classical message via a quantum channel . [ 23 ] [ 24 ]
In the 1970s, techniques for manipulating single-atom quantum states, such as the atom trap and the scanning tunneling microscope , began to be developed, making it possible to isolate single atoms and arrange them in arrays. Prior to these developments, precise control over single quantum systems was not possible, and experiments used coarser, simultaneous control over a large number of quantum systems. [ 2 ] The development of viable single-state manipulation techniques led to increased interest in the field of quantum information and computation.
In the 1980s, interest arose in whether it might be possible to use quantum effects to disprove Einstein's theory of relativity . If it were possible to clone an unknown quantum state, it would be possible to use entangled quantum states to transmit information faster than the speed of light, disproving Einstein's theory. However, the no-cloning theorem showed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory. [ 2 ]
Despite all the excitement and interest over studying isolated quantum systems and trying to find a way to circumvent the theory of relativity, research in quantum information theory became stagnant in the 1980s. However, around the same time another avenue started dabbling into quantum information and computation: Cryptography . In a general sense, cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another. [ 2 ]
Bennett and Brassard developed a communication channel on which it is impossible to eavesdrop without being detected, a way of communicating secretly at long distances using the BB84 quantum cryptographic protocol. [ 25 ] The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of an eavesdropper in a secure communication line will immediately let the two parties trying to communicate know of the presence of the eavesdropper.
With the advent of Alan Turing 's revolutionary ideas of a programmable computer, or Turing machine , he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine. [ 26 ] [ 27 ] This is known as the Church–Turing thesis .
Soon enough, the first computers were made, and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called Moore's law . This 'law' is a projective trend that states that the number of transistors in an integrated circuit doubles every two years. [ 28 ] As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference. This led to the advent of quantum computing, which uses quantum mechanics to design algorithms.
At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem was developed by David Deutsch and Richard Jozsa , known as the Deutsch–Jozsa algorithm . This problem however held little to no practical applications. [ 2 ] Peter Shor in 1994 came up with a very important and practical problem , one of finding the prime factors of an integer. The discrete logarithm problem as it was called, could theoretically be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers should be more powerful than Turing machines.
Around the time computer science was making a revolution, so was information theory and communication, through Claude Shannon . [ 29 ] [ 30 ] [ 31 ] Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and noisy channel coding theorem . He also showed that error correcting codes could be used to protect information being sent.
Quantum information theory also followed a similar trajectory, Ben Schumacher in 1995 made an analogue to Shannon's noiseless coding theorem using the qubit . A theory of error-correction also developed, which allows quantum computers to make efficient computations regardless of noise and make reliable communication over noisy quantum channels. [ 2 ]
Quantum information differs strongly from classical information, epitomized by the bit , in many striking and unfamiliar ways. While the fundamental unit of classical information is the bit , the most basic unit of quantum information is the qubit . Classical information is measured using Shannon entropy , while the quantum mechanical analogue is Von Neumann entropy . Given a statistical ensemble of quantum mechanical systems with the density matrix ρ {\displaystyle \rho } , it is given by S ( ρ ) = − Tr ( ρ ln ρ ) . {\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).} [ 2 ] Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy [ 32 ] and the conditional quantum entropy .
Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the Bloch sphere . Despite being continuously valued in this way, a qubit is the smallest possible unit of quantum information, and despite the qubit state being continuous-valued, it is impossible to measure the value precisely. Five famous theorems describe the limits on manipulation of quantum information. [ 2 ]
These theorems are proven from unitarity , which according to Leonard Susskind is the technical term for the statement that quantum information within the universe is conserved. [ 33 ] : 94 The five theorems open possibilities in quantum information processing.
The state of a qubit contains all of its information. This state is frequently expressed as a vector on the Bloch sphere. This state can be changed by applying linear transformations or quantum gates to them. These unitary transformations are described as rotations on the Bloch sphere. While classical gates correspond to the familiar operations of Boolean logic , quantum gates are physical unitary operators .
The study of the above topics and differences comprises quantum information theory.
Quantum mechanics is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a photon in a linear optical quantum computer , an ion in a trapped ion quantum computer , or it might be a large collection of atoms as in a superconducting quantum computer . Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers . Another important difference with quantum mechanics is that while quantum mechanics often studies infinite-dimensional systems such as a harmonic oscillator , quantum information theory is concerned with both continuous-variable systems [ 34 ] and finite-dimensional systems. [ 8 ] [ 35 ] [ 36 ]
Entropy measures the uncertainty in the state of a physical system. [ 2 ] Entropy can be studied from the point of view of both the classical and quantum information theories.
Classical information is based on the concepts of information laid out by Claude Shannon . Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit. [ 37 ]
Shannon entropy is the quantification of the information gained by measuring the value of a random variable. Another way of thinking about it is by looking at the uncertainty of a system prior to measurement. As a result, entropy, as pictured by Shannon, can be seen either as a measure of the uncertainty prior to making a measurement or as a measure of information gained after making said measurement. [ 2 ]
Shannon entropy, written as a function of a discrete probability distribution, P ( x 1 ) , P ( x 2 ) , . . . , P ( x n ) {\displaystyle P(x_{1}),P(x_{2}),...,P(x_{n})} associated with events x 1 , . . . , x n {\displaystyle x_{1},...,x_{n}} , can be seen as the average information associated with this set of events, in units of bits:
H ( X ) = H [ P ( x 1 ) , P ( x 2 ) , . . . , P ( x n ) ] = − ∑ i = 1 n P ( x i ) log 2 P ( x i ) {\displaystyle H(X)=H[P(x_{1}),P(x_{2}),...,P(x_{n})]=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})}
This definition of entropy can be used to quantify the physical resources required to store the output of an information source. The ways of interpreting Shannon entropy discussed above are usually only meaningful when the number of samples of an experiment is large. [ 35 ]
The Rényi entropy is a generalization of Shannon entropy defined above. The Rényi entropy of order r, written as a function of a discrete probability distribution, P ( a 1 ) , P ( a 2 ) , . . . , P ( a n ) {\displaystyle P(a_{1}),P(a_{2}),...,P(a_{n})} , associated with events a 1 , . . . , a n {\displaystyle a_{1},...,a_{n}} , is defined as: [ 37 ]
H r ( A ) = 1 1 − r log 2 ∑ i = 1 n P r ( a i ) {\displaystyle H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})}
for 0 < r < ∞ {\displaystyle 0<r<\infty } and r ≠ 1 {\displaystyle r\neq 1} .
We arrive at the definition of Shannon entropy from Rényi when r → 1 {\displaystyle r\rightarrow 1} , of Hartley entropy (or max-entropy) when r → 0 {\displaystyle r\rightarrow 0} , and min-entropy when r → ∞ {\displaystyle r\rightarrow \infty } .
Quantum information theory is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made. [ 37 ]
One interpretation of Shannon entropy was the uncertainty associated with a probability distribution. When we want to describe the information or the uncertainty of a quantum state, the probability distributions are simply replaced by density operators ρ {\displaystyle \rho } :
S ( ρ ) ≡ − t r ( ρ log 2 ρ ) = − ∑ i λ i log 2 λ i , {\displaystyle S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},}
where λ i {\displaystyle \lambda _{i}} are the eigenvalues of ρ {\displaystyle \rho } .
Von Neumann entropy plays a role in quantum information similar to the role Shannon entropy plays in classical information.
Quantum communication is one of the applications of quantum physics and quantum information. There are some famous theorems such as the no-cloning theorem that illustrate some important properties in quantum communication. Dense coding and quantum teleportation are also applications of quantum communication. They are two opposite ways to communicate using qubits. While teleportation transfers one qubit from Alice and Bob by communicating two classical bits under the assumption that Alice and Bob have a pre-shared Bell state , dense coding transfers two classical bits from Alice to Bob by using one qubit, again under the same assumption, that Alice and Bob have a pre-shared Bell state.
One of the best known applications of quantum cryptography is quantum key distribution which provide a theoretical solution to the security issue of a classical key. The advantage of quantum key distribution is that it is impossible to copy a quantum key because of the no-cloning theorem . If someone tries to read encoded data, the quantum state being transmitted will change. This could be used to detect eavesdropping.
The first quantum key distribution scheme, BB84 , was developed by Charles Bennett and Gilles Brassard in 1984. It is usually explained as a method of securely communicating a private key from a third party to another for use in one-time pad encryption. [ 2 ]
E91 was made by Artur Ekert in 1991. His scheme uses entangled pairs of photons. These two photons can be created by Alice, Bob, or by a third party including eavesdropper Eve. One of the photons is distributed to Alice and the other to Bob so that each one ends up with one photon from the pair.
This scheme relies on two properties of quantum entanglement:
B92 is a simpler version of BB84. [ 38 ]
The main difference between B92 and BB84:
Like the BB84, Alice transmits to Bob a string of photons encoded with randomly chosen bits but this time the bits Alice chooses the bases she must use. Bob still randomly chooses a basis by which to measure but if he chooses the wrong basis, he will not measure anything which is guaranteed by quantum mechanics theories. Bob can simply tell Alice after each bit she sends whether he measured it correctly. [ 39 ]
The most widely used model in quantum computation is the quantum circuit , which are based on the quantum bit " qubit ". Qubit is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state , or they can be in a superposition of the 1 and 0 states. However, when qubits are measured, the result of the measurement is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that the qubits were in immediately prior to the measurement.
Any quantum computation algorithm can be represented as a network of quantum logic gates .
If a quantum system were perfectly isolated, it would maintain coherence perfectly, but it would be impossible to test the entire system. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; this process is called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics.
QEC is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise . Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.
Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of ancilla qubits . A quantum error correcting code protects quantum information against errors.
Many journals publish research in quantum information science , although only a few are dedicated to this area. Among these are: | https://en.wikipedia.org/wiki/Quantum_information |
In quantum physics , a quantum instrument is a mathematical description of a quantum measurement, capturing both the classical and quantum outputs. [ 1 ] It can be equivalently understood as a quantum channel that takes as input a quantum system and has as its output two systems: a classical system containing the outcome of the measurement and a quantum system containing the post-measurement state. [ 2 ]
Let X {\displaystyle X} be a countable set describing the outcomes of a quantum measurement, and let { E x } x ∈ X {\displaystyle \{{\mathcal {E}}_{x}\}_{x\in X}} denote a collection of trace-non-increasing completely positive maps , such that the sum of all E x {\displaystyle {\mathcal {E}}_{x}} is trace-preserving, i.e. tr ( ∑ x E x ( ρ ) ) = tr ( ρ ) {\textstyle \operatorname {tr} \left(\sum _{x}{\mathcal {E}}_{x}(\rho )\right)=\operatorname {tr} (\rho )} for all positive operators ρ . {\displaystyle \rho .}
Now for describing a measurement by an instrument I {\displaystyle {\mathcal {I}}} , the maps E x {\displaystyle {\mathcal {E}}_{x}} are used to model the mapping from an input state ρ {\displaystyle \rho } to the output state of a measurement conditioned on a classical measurement outcome x {\displaystyle x} . Therefore, the probability that a specific measurement outcome x {\displaystyle x} occurs on a state ρ {\displaystyle \rho } is given by [ 3 ] [ 4 ] p ( x | ρ ) = tr ( E x ( ρ ) ) . {\displaystyle p(x|\rho )=\operatorname {tr} ({\mathcal {E}}_{x}(\rho )).}
The state after a measurement with the specific outcome x {\displaystyle x} is given by [ 3 ] [ 4 ]
ρ x = E x ( ρ ) tr ( E x ( ρ ) ) . {\displaystyle \rho _{x}={\frac {{\mathcal {E}}_{x}(\rho )}{\operatorname {tr} ({\mathcal {E}}_{x}(\rho ))}}.}
If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections | x ⟩ ⟨ x | ∈ B ( C | X | ) {\textstyle |x\rangle \langle x|\in {\mathcal {B}}(\mathbb {C} ^{|X|})} , then the action of an instrument I {\displaystyle {\mathcal {I}}} is given by a quantum channel I : B ( H 1 ) → B ( H 2 ) ⊗ B ( C | X | ) {\displaystyle {\mathcal {I}}:{\mathcal {B}}({\mathcal {H}}_{1})\rightarrow {\mathcal {B}}({\mathcal {H}}_{2})\otimes {\mathcal {B}}(\mathbb {C} ^{|X|})} with [ 2 ]
I ( ρ ) := ∑ x E x ( ρ ) ⊗ | x ⟩ ⟨ x | . {\displaystyle {\mathcal {I}}(\rho ):=\sum _{x}{\mathcal {E}}_{x}(\rho )\otimes \vert x\rangle \langle x|.}
Here H 1 {\displaystyle {\mathcal {H}}_{1}} and H 2 ⊗ C | X | {\displaystyle {\mathcal {H}}_{2}\otimes \mathbb {C} ^{|X|}} are the Hilbert spaces corresponding to the input and the output systems of the instrument.
Just as a completely positive trace preserving (CPTP) map can always be considered as the reduction of unitary evolution on a system with an initially unentangled auxiliary, quantum instruments are the reductions of projective measurement with a conditional unitary, and also reduce to CPTP maps and POVMs when ignore measurement outcomes and state evolution, respectively. [ 4 ] In John Smolin 's terminology, this is an example of "going to the Church of the Larger Hilbert space ".
Any quantum instrument on a system S {\displaystyle {\mathcal {S}}} can be modeled as a projective measurement on S {\displaystyle {\mathcal {S}}} and (jointly) an uncorrelated auxiliary A {\displaystyle {\mathcal {A}}} followed by a unitary conditional on the measurement outcome. [ 3 ] [ 4 ] Let η {\displaystyle \eta } (with η > 0 {\displaystyle \eta >0} and T r η = 1 {\displaystyle \mathrm {Tr} \,\eta =1} ) be the normalized initial state of A {\displaystyle {\mathcal {A}}} , let { Π i } {\displaystyle \{\Pi _{i}\}} (with Π i = Π i † = Π i 2 {\displaystyle \Pi _{i}=\Pi _{i}^{\dagger }=\Pi _{i}^{2}} and Π i Π j = δ i j Π i {\displaystyle \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}} ) be a projective measurement on S A {\displaystyle {\mathcal {SA}}} , and let { U i } {\displaystyle \{U_{i}\}} (with U i † = U i − 1 {\displaystyle U_{i}^{\dagger }=U_{i}^{-1}} ) be unitaries on S A {\displaystyle {\mathcal {SA}}} . Then one can check that
defines a quantum instrument. [ 4 ] Furthermore, one can also check that any choice of quantum instrument { E i } {\displaystyle \{{\mathcal {E}}_{i}\}} can be obtained with this construction for some choice of η {\displaystyle \eta } and { U i } {\displaystyle \{U_{i}\}} . [ 4 ]
In this sense, a quantum instrument can be thought of as the reduction of a projective measurement combined with a conditional unitary.
Any quantum instrument { E i } {\displaystyle \{{\mathcal {E}}_{i}\}} immediately induces a CPTP map, i.e., a quantum channel: [ 4 ]
This can be thought of as the overall effect of the measurement on the quantum system if the measurement outcome is thrown away.
Any quantum instrument { E i } {\displaystyle \{{\mathcal {E}}_{i}\}} immediately induces a positive operator-valued measurement ( POVM ):
where K a ( i ) {\displaystyle K_{a}^{(i)}} are any choice of Kraus operators for E i {\displaystyle {\mathcal {E}}_{i}} , [ 4 ]
The Kraus operators K a ( i ) {\displaystyle K_{a}^{(i)}} are not uniquely determined by the CP maps E i {\displaystyle {\mathcal {E}}_{i}} , but the above definition of the POVM elements M i {\displaystyle M_{i}} is the same for any choice. [ 4 ] The POVM can be thought of as the measurement of the quantum system if the information about how the system is affected by the measurement is thrown away. | https://en.wikipedia.org/wiki/Quantum_instrument |
In quantum physics , the quantum inverse scattering method (QISM), similar to the closely related algebraic Bethe ansatz , is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979. [ 1 ]
It can be viewed as a quantized version of the classical inverse scattering method pioneered by Norman Zabusky and Martin Kruskal [ 2 ] used to investigate the Korteweg–de Vries equation and later other integrable partial differential equations . In both, a Lax matrix features heavily and scattering data is used to construct solutions to the original system.
While the classical inverse scattering method is used to solve integrable partial differential equations which model continuous media (for example, the KdV equation models shallow water waves), the QISM is used to solve many-body quantum systems, sometimes known as spin chains , of which the Heisenberg spin chain is the best-studied and most famous example. These are typically discrete systems, with particles fixed at different points of a lattice, but limits of results obtained by the QISM can give predictions even for field theories defined on a continuum, such as the quantum sine-Gordon model .
The quantum inverse scattering method relates two different approaches:
This method led to the formulation of quantum groups , in particular the Yangian . [ citation needed ] The center of the Yangian, given by the quantum determinant plays a prominent role in the method. [ citation needed ]
An important concept in the inverse scattering transform is the Lax representation . The quantum inverse scattering method starts by the quantization of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the algebraic Bethe ansatz . [ 3 ] This led to further progress in the understanding of quantum integrable systems , such as the quantum Heisenberg model , the quantum nonlinear Schrödinger equation (also known as the Lieb–Liniger model or the Tonks–Girardeau gas ) and the Hubbard model . [ citation needed ]
The theory of correlation functions was developed [ when? ] , relating determinant representations, descriptions by differential equations and the Riemann–Hilbert problem . Asymptotics of correlation functions which include space, time and temperature dependence were evaluated in 1991. [ citation needed ]
Explicit expressions for the higher conservation laws of the integrable models were obtained in 1989. [ citation needed ]
Essential progress was achieved in study of ice-type models : the bulk free energy of the
six vertex model depends on boundary conditions even in the thermodynamic limit . [ citation needed ]
The steps can be summarized as follows (Evgeny Sklyanin 1992 ): | https://en.wikipedia.org/wiki/Quantum_inverse_scattering_method |
A quantum jump is the abrupt transition of a quantum system ( atom , molecule , atomic nucleus ) from one quantum state to another, from one energy level to another. When the system absorbs energy, there is a transition to a higher energy level ( excitation ); when the system loses energy, there is a transition to a lower energy level.
The concept was introduced by Niels Bohr , in his 1913 Bohr model .
A quantum jump is a phenomenon that is peculiar to quantum systems and distinguishes them from classical systems, where any transitions are performed gradually. In quantum mechanics, such jumps are associated with the non-unitary evolution of a quantum-mechanical system during measurement.
A quantum jump can be accompanied by the emission or absorption of photons ; energy transfer during a quantum jump can also occur by non-radiative resonant energy transfer or in collisions with other particles.
In modern physics, the concept of a quantum jump is rarely used; as a rule scientists speak of transitions between quantum states or energy levels.
Atomic electron transitions cause the emission or absorption of photons . Their statistics are Poissonian , and the time between jumps is exponentially distributed . [ 1 ] The damping time constant (which ranges from nanoseconds to a few seconds) relates to the natural, pressure, and field broadening of spectral lines . The larger the energy separation of the states between which the electron jumps, the shorter the wavelength of the photon emitted.
In an ion trap , quantum jumps can be directly observed by addressing a trapped ion with radiation at two different frequencies to drive electron transitions. [ 2 ] This requires one strong and one weak transition to be excited (denoted ω {\displaystyle \omega } 12 and ω {\displaystyle \omega } 13 respectively in the figure to the right). The electron energy level, | 2 ⟩ {\displaystyle |2\rangle } , has a short lifetime, Γ {\displaystyle \Gamma } 2 which allows for constant emission of photons at a frequency ω {\displaystyle \omega } 12 which can be collected by a camera and/or photomultiplier tube . State | 3 ⟩ {\displaystyle |3\rangle } has a relatively long lifetime Γ {\displaystyle \Gamma } 3 which causes an interruption of the photon emission as the electron gets shelved in state through application of light with frequency ω {\displaystyle \omega } 13. The ion going dark is a direct observation of quantum jumps. | https://en.wikipedia.org/wiki/Quantum_jump |
The quantum jump method , also known as the Monte Carlo wave function (MCWF) is a technique in computational physics used for simulating open quantum systems and quantum dissipation . The quantum jump method was developed by Dalibard , Castin and Mølmer at a similar time to the similar method known as Quantum Trajectory Theory developed by Carmichael . Other contemporaneous works on wave-function-based Monte Carlo approaches to open quantum systems include those of Dum, Zoller and Ritsch and Hegerfeldt and Wilser. [ 1 ] [ 2 ]
The quantum jump method is an approach which is much like the master-equation treatment except that it operates on the wave function rather than using a density matrix approach. The main component of this method is evolving the system's wave function in time with a pseudo-Hamiltonian; where at each time step, a quantum jump (discontinuous change) may take place with some probability. The calculated system state as a function of time is known as a quantum trajectory , and the desired density matrix as a function of time may be calculated by averaging over many simulated trajectories. For a Hilbert space of dimension N, the number of wave function components is equal to N while the number of density matrix components is equal to N 2 . Consequently, for certain problems the quantum jump method offers a performance advantage over direct master-equation approaches. [ 1 ]
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_jump_method |
A quantum limit in physics is a limit on measurement accuracy at quantum scales. [ 1 ] Depending on the context, the limit may be absolute (such as the Heisenberg limit ), or it may only apply when the experiment is conducted with naturally occurring quantum states (e.g. the standard quantum limit in interferometry) and can be circumvented with advanced state preparation and measurement schemes.
The usage of the term standard quantum limit or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical observable of a system under study that does not commute with itself at different times leads to such limits. In short, it is the Heisenberg uncertainty principle that is the cause.
A more detailed explanation would be that any measurement in quantum mechanics involves at least two parties, an Object and a Meter. The former is the system whose observable, say x ^ {\displaystyle {\hat {x}}} , we want to measure. The latter is the system we couple to the Object in order to infer the value of x ^ {\displaystyle {\hat {x}}} of the Object by recording some chosen observable, O ^ {\displaystyle {\hat {\mathcal {O}}}} , of this system, e.g. the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as indirect measurements (see pp. 38–42 of [ 1 ] ). So any measurement is a result of interaction and that acts in both ways. Therefore, the Meter acts on the Object during each measurement, usually via the quantity, F ^ {\displaystyle {\hat {\mathcal {F}}}} , conjugate to the readout observable O ^ {\displaystyle {\hat {\mathcal {O}}}} , thus perturbing the value of measured observable x ^ {\displaystyle {\hat {x}}} and modifying the results of subsequent measurements. This is known as back action (quantum) of the Meter on the system under measurement.
At the same time, quantum mechanics prescribes that readout observable of the Meter should have an inherent uncertainty, δ O ^ {\displaystyle \delta {\hat {\mathcal {O}}}} , additive to and independent of the value of the measured quantity x ^ {\displaystyle {\hat {x}}} . This one is known as measurement imprecision or measurement noise . Because of the Heisenberg uncertainty principle , this imprecision cannot be arbitrary and is linked to the back-action perturbation by the uncertainty relation :
where Δ a = ⟨ a ^ 2 ⟩ − ⟨ a ^ ⟩ 2 {\displaystyle \Delta a={\sqrt {\langle {\hat {a}}^{2}\rangle -\langle {\hat {a}}\rangle ^{2}}}} is a standard deviation of observable a {\displaystyle a} and ⟨ a ^ ⟩ {\displaystyle \langle {\hat {a}}\rangle } stands for expectation value of a {\displaystyle a} in whatever quantum state the system is. The equality is reached if the system is in a minimum uncertainty state . The consequence for our case is that the more precise is our measurement, i.e the smaller is Δ δ O {\displaystyle \Delta {\mathcal {\delta O}}} , the larger will be perturbation the Meter exerts on the measured observable x ^ {\displaystyle {\hat {x}}} . Therefore, the readout of the meter will, in general, consist of three terms:
where x ^ f r e e {\displaystyle {\hat {x}}_{\mathrm {free} }} is a value of x ^ {\displaystyle {\hat {x}}} that the Object would have, were it not coupled to the Meter, and δ x B A ^ [ F ^ ] {\displaystyle \delta {\hat {x_{BA}}}[{\hat {\mathcal {F}}}]} is the perturbation to the value of x ^ {\displaystyle {\hat {x}}} caused by back action force, F ^ {\displaystyle {\hat {\mathcal {F}}}} . The uncertainty of the latter is proportional to Δ F ∝ Δ O − 1 {\displaystyle \Delta {\mathcal {F}}\propto \Delta {\mathcal {O}}^{-1}} . Thus, there is a minimal value, or the limit to the precision one can get in such a measurement, provided that δ O ^ {\displaystyle \delta {\hat {\mathcal {O}}}} and F ^ {\displaystyle {\hat {\mathcal {F}}}} are uncorrelated. [ 2 ] [ 3 ]
The terms "quantum limit" and "standard quantum limit" are sometimes used interchangeably. Usually, "quantum limit" is a general term which refers to any restriction on measurement due to quantum effects, while the "standard quantum limit" in any given context refers to a quantum limit which is ubiquitous in that context.
Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body M {\displaystyle M} . The position x {\displaystyle x} of M {\displaystyle M} is probed periodically with time interval ϑ {\displaystyle \vartheta } . We assume mass M {\displaystyle M} large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.
Then each j {\displaystyle j} -th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position x ( t j ) {\displaystyle x(t_{j})} at the moment of reflection:
where k p = ω p / c {\displaystyle k_{p}=\omega _{p}/c} , ω p {\displaystyle \omega _{p}} is the light frequency, j = … , − 1 , 0 , 1 , … {\displaystyle j=\dots ,-1,0,1,\dots } is the pulse number and ϕ ^ j {\displaystyle {\hat {\phi }}_{j}} is the initial (random) phase of the j {\displaystyle j} -th pulse. We assume that the mean value of all these phases is equal to zero, ⟨ ϕ ^ j ⟩ = 0 {\displaystyle \langle {\hat {\phi }}_{j}\rangle =0} , and their root mean square (RMS) uncertainty ( ⟨ ϕ 2 ^ ⟩ − ⟨ ϕ ^ ⟩ 2 ) 1 / 2 {\displaystyle (\langle {\hat {\phi ^{2}}}\rangle -\langle {\hat {\phi }}\rangle ^{2})^{1/2}} is equal to Δ ϕ {\displaystyle \Delta \phi } .
The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector can be done using e.g. homodyne or heterodyne detection schemes (see Section 2.3 in [ 2 ] and references therein), or other such read-out techniques.
In this example, light pulse phase ϕ ^ j {\displaystyle {\hat {\phi }}_{j}} serves as the readout observable O {\displaystyle {\mathcal {O}}} of the Meter. Then we suppose that the phase ϕ ^ j r e f l {\displaystyle {\hat {\phi }}_{j}^{\mathrm {refl} }} measurement error introduced by the detector is much smaller than the initial uncertainty of the phases Δ ϕ {\displaystyle \Delta \phi } . In this case, the initial uncertainty will be the only source of the position measurement error:
For convenience, we renormalise Eq. ( 1 ) as the equivalent test-mass displacement:
where
are the independent random values with the RMS uncertainties given by Eq. ( 2 ).
Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to
where p ^ j b e f o r e {\displaystyle {\hat {p}}_{j}^{\mathrm {before} }} and p ^ j a f t e r {\displaystyle {\hat {p}}_{j}^{\mathrm {after} }} are the test-mass momentum values just before and just after the light pulse reflection, and W j {\displaystyle {\mathcal {W}}_{j}} is the energy of the j {\displaystyle j} -th pulse, that plays the role of back action observable F ^ {\displaystyle {\hat {\mathcal {F}}}} of the Meter. The major part of this perturbation is contributed by classical radiation pressure:
with W {\displaystyle {\mathcal {W}}} the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator. The random part, which cannot be compensated, is proportional to the deviation of the pulse energy:
and its RMS uncertainly is equal to
with Δ W {\displaystyle \Delta {\mathcal {W}}} the RMS uncertainty of the pulse energy.
Assuming the mirror is free (which is a fair approximation if time interval between pulses is much shorter than the period of suspended mirror oscillations, ϑ ≪ T {\displaystyle \vartheta \ll T} ), one can estimate an additional displacement caused by the back action of the j {\displaystyle j} -th pulse that will contribute to the uncertainty of the subsequent measurement by the j + 1 {\displaystyle j+1} pulse time ϑ {\displaystyle \vartheta } later:
Its uncertainty will be simply
If we now want to estimate how much has the mirror moved between the j {\displaystyle j} and j + 1 {\displaystyle j+1} pulses, i.e. its displacement δ x ~ j + 1 , j = x ~ ( t j + 1 ) − x ~ ( t j ) {\displaystyle \delta {\tilde {x}}_{j+1,j}={\tilde {x}}(t_{j+1})-{\tilde {x}}(t_{j})} , we will have to deal with three additional uncertainties that limit precision of our estimate:
where we assumed all contributions to our measurement uncertainty statistically independent and thus got sum uncertainty by summation of standard deviations. If we further assume that all light pulses are similar and have the same phase uncertainty, thence Δ x m e a s ( t j + 1 ) = Δ x m e a s ( t j ) ≡ Δ x m e a s = Δ ϕ / ( 2 k p ) {\displaystyle \Delta x_{\rm {meas}}(t_{j+1})=\Delta x_{\rm {meas}}(t_{j})\equiv \Delta x_{\rm {meas}}=\Delta \phi /(2k_{p})} .
Now, what is the minimum this sum and what is the minimum error one can get in this simple estimate? The answer ensues from quantum mechanics, if we recall that energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:
Therefore, it follows from Eqs. ( 2 and 5 ) that the position measurement error Δ x m e a s {\displaystyle \Delta x_{\mathrm {meas} }} and the momentum perturbation Δ p b . a . {\displaystyle \Delta p_{\mathrm {b.a.} }} due to back action also satisfy the uncertainty relation:
Taking this relation into account, the minimal uncertainty, Δ x m e a s {\displaystyle \Delta x_{\mathrm {meas} }} , the light pulse should have in order not to perturb the mirror too much, should be equal to Δ x b . a . {\displaystyle \Delta x_{\mathrm {b.a.} }} yielding for both Δ x m i n = ℏ ϑ 2 M {\displaystyle \Delta x_{\mathrm {min} }={\sqrt {\frac {\hbar \vartheta }{2M}}}} . Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:
This is the Standard Quantum Limit for such a 2-pulse procedure. In principle, if we limit our measurement to two pulses only and do not care about perturbing mirror position afterwards, the second pulse measurement uncertainty, Δ x m e a s ( t j + 1 ) {\displaystyle \Delta x_{\rm {meas}}(t_{j+1})} , can, in theory, be reduced to 0 (it will yield, of course, Δ p b . a . ( t j + 1 ) → ∞ {\displaystyle \Delta p_{\rm {b.a.}}(t_{j+1})\to \infty } ) and the limit of displacement measurement error will reduce to:
which is known as the Standard Quantum Limit for the measurement of free mass displacement.
This example represents a simple particular case of a linear measurement . This class of measurement schemes can be fully described by two linear equations of the form~( 3 ) and ( 4 ), provided that both the measurement uncertainty and the object back-action perturbation ( x ^ f l ( t j ) {\displaystyle {\hat {x}}_{\mathrm {fl} }(t_{j})} and p ^ b . a . ( t j ) {\displaystyle {\hat {p}}^{\mathrm {b.a.} }(t_{j})} in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case).
In the context of interferometry or other optical measurements, the standard quantum limit usually refers to the minimum level of quantum noise which is obtainable without squeezed states . [ 4 ]
There is additionally a quantum limit for phase noise , reachable only by a laser at high noise frequencies.
In spectroscopy , the shortest wavelength in an X-ray spectrum is called the quantum limit. [ 5 ]
Note that due to an overloading of the word "limit", the classical limit is not the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the Armstrong limit ). In "classical limit", "limit" is used in the sense of a limiting process . (Note that there is no simple rigorous mathematical limit which fully recovers classical mechanics from quantum mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantum_limit |
Quantum logic spectroscopy ( QLS ) is an ion control scheme that maps quantum information between two co-trapped ion species. [ 1 ] Quantum logic operations allow desirable properties of each ion species to be utilized simultaneously. This enables work with ions and molecular ions that have complex internal energy level structures which preclude laser cooling and direct manipulation of state. QLS was first demonstrated by NIST in 2005. [ 1 ] QLS was first applied to state detection in diatomic molecules in 2016 by Wolf et al, [ 2 ] and later applied to state manipulation and detection of diatomic molecules by the Liebfried group at NIST in 2017 [ 3 ]
Lasers are used to couple each ion's internal and external motional degrees of freedom. The Coulomb interaction between the two ions couples their motion. This allows the internal state of one ion to be transferred to the other. An auxiliary "logic ion" provides cooling, state preparation, and state detection for the co-trapped "spectroscopy ion," which has an electronic transition of interest. The logic ion is used to sense and control the internal and external state of the spectroscopy ion. [ 4 ] [ 5 ] [ 6 ]
The logic ion is selected to have a simple energy level structure that can be directly laser cooled, often an alkaline earth ion. The laser cooled logic ion provides sympathetic cooling to the spectroscopy ion, which lacks an efficient laser cooling scheme. Cooling the spectroscopy ion reduces the number of rotational and vibrational states that it can occupy. The remaining states are then accessed by driving stimulated Raman spectroscopy transitions with a laser. The light used for driving these transitions is far off-resonant from any electronic transitions. This enables control over the spectroscopy ion's rotational and vibrational state. [ 4 ] [ 5 ] [ 6 ]
Thus far, QLS is limited to diatomic molecules with a mass within 1 AMU of the laser cooled "logic" ion. This is largely due to poorer coupling of the motional states of the occupants of the ion trap as the mass mismatch becomes larger. [ 7 ] Other techniques more tolerant of large mass mismatches are better suited to cases where the ultimate resolution of QLS is not needed, but single-molecule sensitivity is still desired.
The internal states of each ion can be treated as a two level system, with eigenstates denoted | ↑ ⟩ , {\displaystyle |\uparrow \ \rangle ,} and | ↓ ⟩ {\displaystyle |\downarrow \ \rangle } . One of the ion's normal modes is chosen to be the transfer mode used for state mapping. This motional mode must be shared by both ions, which requires both ions be similar in mass. The normal mode has harmonic oscillator states denoted as | n ⟩ m {\displaystyle |n\rangle _{m}} , where n is the nth level of mode m. The wave function
denotes both ions and the transfer mode in the ground state. [ 8 ] S and L represent the spectroscopy and logic ion. The spectroscopy ion's spectroscopy transition is then excited with a laser, producing the state:
A red sideband pi-pulse is then driven on the spectroscopy ion, resulting in the state:
At this stage, the spectroscopy ion's internal state has been mapped on to the transfer mode. The internal state of the ion has been coupled to its motional mode. The | ↓ ⟩ S | 0 ⟩ m {\displaystyle |\downarrow \ \rangle _{S}|0\rangle _{m}} state is unaffected by the pulse of light carrying out this operation because the state | ↑ ⟩ S | − 1 ⟩ m {\displaystyle |\uparrow \ \rangle _{S}|-1\rangle _{m}} does not exist. [ 9 ] [ 6 ] QLS takes advantage of this in order to map the spectroscopy ion's state onto the transfer mode. A final red sideband pi-pulse is applied to the logic ion, resulting in the state:
The spectroscopy ion's initial state has been mapped onto the logic ion, which can then be detected. | https://en.wikipedia.org/wiki/Quantum_logic_spectroscopy |
A quantum machine is a human-made device whose collective motion follows the laws of quantum mechanics . The idea that macroscopic objects may follow the laws of quantum mechanics dates back to the advent of quantum mechanics in the early 20th century. [ 1 ] [ 2 ] However, as highlighted by the Schrödinger's cat thought experiment , quantum effects are not readily observable in large-scale objects. Consequently, quantum states of motion have only been observed in special circumstances at extremely low temperatures. The fragility of quantum effects in macroscopic objects may arise from rapid quantum decoherence . [ 3 ] Researchers created the first quantum machine in 2009, and the achievement was named the "Breakthrough of the Year" by Science in 2010.
The first quantum machine was created on August 4, 2009, by Aaron D. O'Connell while pursuing his Ph.D. under the direction of Andrew N. Cleland and John M. Martinis at the University of California, Santa Barbara . O'Connell and his colleagues coupled together a mechanical resonator , similar to a tiny springboard, and a qubit , a device that can be in a superposition of two quantum states at the same time. They were able to make the resonator vibrate a small amount and a large amount simultaneously—an effect which would be impossible in classical physics . The mechanical resonator was just large enough to see with the naked eye—about as long as the width of a human hair. [ 4 ] The work was subsequently published in the journal Nature in March 2010. [ 5 ] The journal Science declared the creation of the first quantum machine to be the " Breakthrough of the Year " of 2010. [ 6 ]
In order to demonstrate the quantum mechanical behavior, the team first needed to cool the mechanical resonator until it was in its quantum ground state , the state with the lowest possible energy .
A temperature T ≪ h f k {\displaystyle T\ll {\frac {hf}{k}}} was required, where h {\displaystyle h} is the Planck constant , f {\displaystyle f} is the frequency of the resonator, and k {\displaystyle k} is the Boltzmann constant . [a]
Previous teams of researchers had struggled with this stage, as a 1 MHz resonator, for example, would need to be cooled to the extremely low temperature of 50 μK . [ 7 ] O'Connell's team constructed a different type of resonator, a film bulk acoustic resonator , [ 5 ] with a much higher resonant frequency (6 GHz) which would hence reach its ground state at a (relatively) higher temperature (~0.1 K); this temperature could then be easily reached with a dilution refrigerator . [ 5 ] In the experiment, the resonator was cooled to 25 mK. [ 5 ]
The film bulk acoustic resonator was made of piezoelectric material , so that as it oscillated its changing shape created a changing electric signal, and conversely an electric signal could affect its oscillations. This property enabled the resonator to be coupled with a superconducting phase qubit , a device used in quantum computing whose quantum state can be accurately controlled.
In quantum mechanics, vibrations are made up of elementary vibrations called phonons . Cooling the resonator to its ground state can be seen as equivalent to removing all of the phonons. The team was then able to transfer individual phonons from the qubit to the resonator. The team was also able to transfer a superposition state, where the qubit was in a superposition of two states at the same time, onto the mechanical resonator. [ 8 ] This means the resonator "literally vibrated a little and a lot at the same time", according to the American Association for the Advancement of Science . [ 9 ] The vibrations lasted just a few nanoseconds before being broken down by disruptive outside influences. [ 10 ] In the Nature paper, the team concluded "This demonstration provides strong evidence that quantum mechanics applies to a mechanical object large enough to be seen with the naked eye." [ 5 ]
^ a: The ground state energy of an oscillator is proportional to its frequency: see quantum harmonic oscillator . | https://en.wikipedia.org/wiki/Quantum_machine |
A quantum master equation is a generalization of the idea of a master equation . Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix ), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements . A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.
A formally exact quantum master equation is the Nakajima–Zwanzig equation , which is in general as difficult to solve as the full quantum problem.
The Redfield equation and Lindblad equation are examples of approximate Markovian quantum master equations. These equations are very easy to solve, but are not generally accurate.
Some modern approximations based on quantum master equations, which show better agreement with exact numerical calculations in some cases, include the polaron transformed quantum master equation and the VPQME (variational polaron transformed quantum master equation). [ 1 ]
Numerically exact approaches to the kinds of problems to which master equations are usually applied include numerical Feynman integrals , [ 2 ] quantum Monte Carlo , DMRG [ 3 ] and NRG , MCTDH , [ 4 ] and HEOM . | https://en.wikipedia.org/wiki/Quantum_master_equation |
Quantum materials is an umbrella term in condensed matter physics that encompasses all materials whose essential properties cannot be described in terms of semiclassical particles and low-level quantum mechanics . [ 1 ] These are materials that present strong electronic correlations or some type of electronic order, such as superconducting or magnetic orders, or materials whose electronic properties are linked to non-generic quantum effects – topological insulators , Dirac electron systems such as graphene , as well as systems whose collective properties are governed by genuinely quantum behavior, such as ultra-cold atoms , cold excitons , polaritons , and so forth. On the microscopic level, four fundamental degrees of freedom – that of charge, spin, orbit and lattice – become intertwined, resulting in complex electronic states; [ 1 ] the concept of emergence is a common thread in the study of quantum materials. [ 2 ]
Quantum materials exhibit puzzling properties with no counterpart in the macroscopic world: quantum entanglement, quantum fluctuations, robust boundary states dependent on the topology of the materials' bulk wave functions, etc. [ 1 ] Quantum anomalies such as the chiral magnetic effect link some quantum materials with processes in high-energy physics of quark-gluon plasmas . [ 3 ]
In 2012, Joseph Orenstein published an article in Physics Today about "ultrafast spectroscopy of quantum materials". [ 4 ] Orenstein stated,
Quantum materials is a label that has come to signify the area of condensed-matter physics formerly known as strongly correlated electronic systems. Although the field is broad, a unifying theme is the discovery and investigation of materials whose electronic properties cannot be understood with concepts from contemporary condensed-matter textbooks.
As a paradigmatic example, Orenstein refers to the breakdown of Landau Fermi liquid theory due to strong correlations. The use of the term "quantum materials" has been extended and applied to other systems, such as topological insulators, and Dirac electron materials. The term has gained momentum since the article "The rise of quantum materials" was published in Nature Physics in 2016. [ 2 ] Quoting:
on a trivial level all materials exist thanks to the laws of quantum mechanics, and there are cynics who will privately wonder if the description isn't too broad and, well, catchy for its own good. But given the history of condensed-matter physics that we have just outlined, there are good reasons to embrace quantum materials. In essence, they provide a common thread linking disparate communities of researchers working on a variety of problems at the frontiers of physics, materials science and engineering. | https://en.wikipedia.org/wiki/Quantum_materials |
In pair production , a photon creates an electron positron pair. In the process of photons scattering in air (e.g. in lightning discharges), the most important interaction is the scattering of photons at the nuclei of atoms or molecules . The full quantum mechanical process of pair production can be described by the quadruply differential cross section given here: [ 1 ]
with
This expression can be derived by using a quantum mechanical symmetry between pair production and Bremsstrahlung . Z {\displaystyle Z} is the atomic number , α f i n e ≈ 1 / 137 {\displaystyle \alpha _{fine}\approx 1/137} the fine structure constant , ℏ {\displaystyle \hbar } the reduced Planck constant and c {\displaystyle c} the speed of light . The kinetic energies E k i n , + / − {\displaystyle E_{kin,+/-}} of the positron and electron relate to their total energies E + , − {\displaystyle E_{+,-}} and momenta p + , − {\displaystyle \mathbf {p} _{+,-}} via
Conservation of energy yields
The momentum q {\displaystyle \mathbf {q} } of the virtual photon between incident photon and nucleus is:
where the directions are given via:
where k {\displaystyle \mathbf {k} } is the momentum of the incident photon.
In order to analyse the relation between the photon energy E + {\displaystyle E_{+}} and the emission angle Θ + {\displaystyle \Theta _{+}} between photon and positron, Köhn and Ebert integrated [ 2 ] the quadruply differential cross section over Θ − {\displaystyle \Theta _{-}} and Φ {\displaystyle \Phi } . The double differential cross section is:
with
and
This cross section can be applied in Monte Carlo simulations. An analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon. | https://en.wikipedia.org/wiki/Quantum_mechanical_scattering_of_photon_and_nucleus |
The theoretical study of time travel generally follows the laws of general relativity . Quantum mechanics requires physicists to solve equations describing how probabilities behave along closed timelike curves (CTCs), which are theoretical loops in spacetime that might make it possible to travel through time. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
In the 1980s, Igor Novikov proposed the self-consistency principle . [ 5 ] According to this principle, any changes made by a time traveler in the past must not create historical paradoxes . If a time traveler attempts to change the past, the laws of physics will ensure that events unfold in a way that avoids paradoxes. This means that while a time traveler can influence past events, those influences must ultimately lead to a consistent historical narrative.
However, Novikov's self-consistency principle has been debated in relation to certain interpretations of quantum mechanics. Specifically, it raises questions about how it interacts with fundamental principles such as unitarity and linearity . Unitarity ensures that the total probability of all possible outcomes in a quantum system always sums to 1, preserving the predictability of quantum events. Linearity ensures that quantum evolution preserves superpositions , allowing quantum systems to exist in multiple states simultaneously. [ 6 ]
There are two main approaches to explaining quantum time travel while incorporating Novikov's self-consistency principle . The first approach uses density matrices to describe the probabilities of different outcomes in quantum systems, providing a statistical framework that can accommodate the constraints of CTCs. The second approach involves state vectors, [ 7 ] which describe the quantum state of a system. Both approaches can lead to insights into how time travel might be reconciled with quantum mechanics, although they may introduce concepts that challenge conventional understandings of these theories. [ 8 ] [ 9 ]
In 1991, David Deutsch proposed a method to explain how quantum systems interact with closed timelike curves (CTCs) using time evolution equations. This method aims to address paradoxes like the grandfather paradox , [ 10 ] [ 11 ] which suggests that a time traveler who stops their own birth would create a contradiction. One interpretation of Deutsch's approach is that it allows for self-consistency without necessarily implying the existence of parallel universes .
To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. To describe the combined evolution of both parts over time, he used a unitary operator ( U ). This approach relies on a specific mathematical framework to describe quantum systems. The overall state is represented by combining the density matrices ( ρ ) for both parts using a tensor product (⊗). [ 12 ] While Deutsch's approach does not assume initial correlation between these two parts, this does not inherently break time symmetry. [ 10 ]
Deutsch's proposal uses the following key equation to describe the fixed-point density matrix ( ρCTC ) for the CTC:
ρ CTC = Tr A [ U ( ρ A ⊗ ρ CTC ) U † ] {\displaystyle \rho _{\text{CTC}}={\text{Tr}}_{A}\left[U\left(\rho _{A}\otimes \rho _{\text{CTC}}\right)U^{\dagger }\right]} .
The unitary evolution involving both the CTC and the external subsystem determines the density matrix of the CTC as a fixed point, focusing on its state.
Deutsch's proposal ensures that the CTC returns to a self-consistent state after each loop. However, if a system retains memories after traveling through a CTC, it could create scenarios where it appears to have experienced different possible pasts. [ 13 ]
Furthermore, Deutsch's method may not align with common probability calculations in quantum mechanics unless we consider multiple paths leading to the same outcome. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing randomness ( nondeterminism ). Deutsch suggested using solutions that maximize entropy , aligning with systems' natural tendency to evolve toward higher entropy states.
To calculate the final state outside the CTC, trace operations consider only the external system's state after combining both systems' evolution.
Deutsch's approach has intriguing implications for paradoxes like the grandfather paradox. For instance, if everything except a single qubit travels through a time machine and flips its value according to a specific operator:
Deutsch argues that maximizing von Neumann entropy is relevant in this context. In this scenario, outcomes may mix starting at 0 and ending at 1 or vice versa. While this interpretation can align with many-worlds views of quantum mechanics, it does not necessarily imply branching timelines after interacting with a CTC. [ 14 ]
Researchers have explored Deutsch's ideas further. If feasible, his model might allow computers near a time machine to solve problems beyond classical capabilities; however, debates about CTCs' feasibility continue. [ 15 ] [ 16 ]
Despite its theoretical nature, Deutsch's proposal has faced significant criticism. [ 17 ] For example, Tolksdorf and Verch demonstrated that quantum systems in spacetimes without CTCs can achieve results similar to Deutsch's criterion with any prescribed accuracy. [ 18 ] [ 19 ] This finding challenges claims that quantum simulations of CTCs are related to closed timelike curves as understood in general relativity . Their research also shows that classical systems governed by statistical mechanics could also meet these criteria [ 20 ] without invoking peculiarities attributed solely to quantum mechanics. Consequently, they argue that their findings raise doubts about Deutsch's explanation of his time travel scenario using many-worlds interpretations of quantum physics.
Seth Lloyd proposed an alternative approach to time travel with closed timelike curves (CTCs), based on " post-selection " and path integrals . [ 21 ] Path integrals are a powerful tool in quantum mechanics that involve summing probabilities over all possible ways a system could evolve, including paths that do not strictly follow a single timeline. [ 22 ] Unlike classical approaches, path integrals can accommodate histories involving CTCs, although their application requires careful consideration of quantum mechanics' principles.
He proposes an equation that describes the transformation of the density matrix, which represents the system's state outside the CTC after a time loop:
In this equation:
The transformation relies on the trace operation, which summarizes aspects of the matrix. If this trace term is zero ( Tr [ C ρ i C † ] = 0 {\displaystyle {\text{Tr}}\left[C\rho _{i}C^{\dagger }\right]=0} ), it indicates that the transformation is invalid in that context, but does not directly imply a paradox like the grandfather paradox. Conversely, a non-zero trace suggests a valid transformation leading to a unique solution for the external system's state.
Thus, Lloyd's approach aims to filter out histories that lead to inconsistencies by allowing only those consistent with both initial and final states. This aligns with post-selection , where specific outcomes are considered based on predetermined criteria; however, it does not guarantee that all paradoxical scenarios are eliminated.
Michael Devin (2001) proposed a model that incorporates closed timelike curves (CTCs) into thermodynamics , [ 23 ] suggesting it as a potential way to address the grandfather paradox. [ 24 ] [ 25 ] This model introduces a "noise" factor to account for imperfections in time travel, proposing a framework that could help mitigate paradoxes. In contrast, Carlo Rovelli has argued that thermodynamics inhibits time travel to the past. [ 26 ] | https://en.wikipedia.org/wiki/Quantum_mechanics_of_time_travel |
Quantum metamaterials apply the science of metamaterials and the rules of quantum mechanics to control electromagnetic radiation . In the broad sense, a quantum metamaterial is a metamaterial in which certain quantum properties of the medium must be taken into account and whose behaviour is thus described by both Maxwell's equations and the Schrödinger equation . Its behaviour reflects the existence of both EM waves and matter waves . The constituents can be at nanoscopic or microscopic scales, depending on the frequency range (e.g., optical or microwave). [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
In a more strict approach, a quantum metamaterial should demonstrate coherent quantum dynamics . Such a system is essentially a spatially extended controllable quantum object that allows additional ways of controlling the propagation of electromagnetic waves. [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ]
Quantum metamaterials can be narrowly defined as optical media that: [ 7 ]
Fundamental research in quantum metamaterials creates opportunities for novel investigations in quantum phase transition , new perspectives on adiabatic quantum computation and a route to other quantum technology applications. Such a system is essentially a spatially-extended controllable quantum object that allows additional ways of controlling electromagnetic wave propagation. [ 6 ] [ 7 ]
In other words, quantum metamaterials incorporate quantum coherent states in order to control and manipulate electromagnetic radiation . With these materials, quantum information processing is combined with the science of metamaterials (periodic artificial electromagnetic materials). The unit cells can be imagined to function as qubits that maintain quantum coherence "long enough for the electromagnetic pulse to travel across". The quantum state is achieved through the material's individual cells. As each cell interacts with the propagating electromagnetic pulse, the whole system retains quantum coherence. [ 6 ] [ 7 ]
Several types of metamaterials are being studied. Nanowires can use quantum dots as the unit cells or artificial atoms of the structure, arranged as periodic nanostructures . This material demonstrates a negative index of refraction and effective magnetism and is simple to build. The radiated wavelength of interest is much larger than the constituent diameter. Another type uses periodically arranged cold atom cells, accomplished with ultra-cold gasses. A photonic bandgap can be demonstrated with this structure, along with tunability and control as a quantum system. [ 3 ] Quantum metamaterial prototypes based on superconducting devices with [ 9 ] [ 10 ] and without [ 11 ] Josephson junctions are being actively investigated. Recently a superconducting quantum metamaterial prototype based on flux qubits was realized. [ 12 ] | https://en.wikipedia.org/wiki/Quantum_metamaterial |
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