text stringlengths 128 2.05k |
|---|
[EQUATION] where [MATH] is a mobility parameter describing the stiffness of the interface dynamic and [MATH] the velocity field moving and deforming the phases again. An interface [MATH] can be represented by the [MATH] contour line (iso-surface in 3D). Based on the interface profile, the surface normals can be approxi... |
[EQUATION] and the mean curvature by [EQUATION] cf. Lucas:Aland2012-xe The above description of the physical two-phase system minimises the area of the interface and thus exhibits unwanted self-dynamics for an interface tracking method. In Lucas:Folch1999-yn it is proposed to add the correction term [MATH] |
to the right hand side, since [MATH] depends on the local curvature, canceling out this effect, which is known as the Allen-Cahn law. Hence, the right-hand side becomes |
[EQUATION] The Diffuse-Domain method has been successfully applied to several biological problems. For example for modelling mechanically induced deformation of bones Lucas:Aland2014-ki |
or simulating endocytosis Lucas:Lowengrub2016-ou It is even possible to couple surface and bulk reactions for modelling transport, diffusion and adsorption of any material quantity in Lucas:Teigen2009-yv Two-phase flows with soluble nanoparticles or soluble surfactants have been modelled with the Diffuse-Domain method ... |
Lucas:Wittwer2016-wj Lucas:Wittwer2017-ru to address a patterning problem in murine lung development. 4.4 Lattice Boltzmann method |
The Lattice Boltzmann method (LBM) is a numerical scheme to simulate fluid dynamics. It evolved from the field of cellular automata, more precisely lattice gas cellular automata (LGCA), in the 1980s. The first LGCA that could simulate fluid flow was proposed in 1986, cp. Frisch1986 However, LGCA were facing several pro... |
The LBM is applied in many different areas that study various different types of fluid dynamics for example incompressible, isothermal, non-isothermal, single- and multi-phase flows, etc., as well as biological flows, see frouzakis2011 As we have seen in the previous sections, biological fluid dynamics can be the dynam... |
In contrast to many conventional approaches modeling fluid flow on the macroscopic scale, for example by solving the Navier-Stokes equations, the LBM is a mesoscopic approach. It is also advantageous for parallel computations because of its local dynamics. LBM models the fluid as fictive particles that propagate and co... |
The Boltzmann equation originates from statistical physics and describes the temporal evolution of a probability density distribution function |
[MATH] defining the probability of finding a particle with velocity [MATH] at location [MATH] at time [MATH] In the presence of an external force [MATH] acting on the particles and considering two processes, i.e. propagation of particles and their collision, the temporal evolution of [MATH] is defined by |
[EQUATION] The most commonly used collision term is the single-relaxation-time Bhatnagar-Gross-Krook (BGK) collision operator [EQUATION] |
with [MATH] being the Maxwell-Boltzmann equilibrium distribution function with a characteristic time scale [MATH] The discretized Boltzmann equation is then |
[EQUATION] where [MATH] determines the number of discrete velocities and [MATH] the spatial dimensionality of the system. The discrete LB equation is given by |
[EQUATION] The equation implicates a two step algorithm for the LBM. In the first step referring to the left hand side of ( 12 ), the probability density distribution functions perform a free flight to the next lattice point. In the second step referring to the right hand side of ( 12 ), the collision of the incoming p... |
The equilibrium function is defined by a second-order expansion of the Maxwell equation in terms of low fluid velocity [EQUATION] |
with the fluid velocity [MATH] , the fluid density [MATH] , the speed of sound [MATH] and the weights [MATH] that are given according to the chosen lattice, cf. He1997 The lattices used in LBM are regular and characterized as [MATH] where [MATH] indicates the spatial dimension and [MATH] the number of discrete velociti... |
[EQUATION] The fluid pressure [MATH] is related to the mass density [MATH] via the equation for an ideal gas [MATH] To simulate fluid-structure interactions, the LBM can be combined with the Immersed Boundary method (IBM), see Peskin2002 , that represents elastic structures being immersed in a fluid. A detailed descrip... |
Conclusion In this chapter, we have given an overview of the mathematical modelling of tissue dynamics, growth, and mechanics in the context of morphogenesis. These different aspects of morphogenesis can be modeled either on a microscopic scale, for example by agent based models, or on a macroscopic scale by continuum ... |
# Source: arxiv 1806.04147 # Title: Entropic uncertainty relations for quantum information scrambling # Sections: all # Downloaded: 2026-03-03T05:15:18.336532+00:00 |
Entropic uncertainty relations for quantum information scrambling Abstract How violently do two quantum operators disagree? Different fields of physics feature different measures of incompatibility: (i) In quantum information theory, entropic uncertainty relations constrain measurement outcomes. (ii) In condensed matte... |
How incompatible are two quantum operators, [MATH] and [MATH] Two species of quantum physicist answer with two different measures. Today’s pure quantum information (QI) theorist checks uncertainty relations cast in terms of entropies Everett_57_Relative Hirschman_57_Note Beckner_75_Inequalities Bialynicki_Birula_75_Unc... |
The second species—the condensed-matter or high-energy physicist—studies the following setup: Consider a strongly coupled quantum many-body system. Examples include an interacting spin chain and the boundary dual of a gravitational theory. The Hamiltonian, [MATH] , couples the subsystems and generates the time-evolutio... |
[MATH] Let [MATH] and [MATH] denote Hermitian and/or unitary operators localized on far-apart subsystems. Examples include Pauli operators acting on opposite sides of the spin chain. In the Heisenberg picture, the interactions delocalize [MATH] to |
[MATH] The support of [MATH] comes to overlap the support of [MATH] the operators cease to agree. The out-of-time-ordered correlator (OTOC) quantifies this disagreement, as well as quantum chaos and scrambling LarkinO_69 Lashkari_13_Towards Kitaev_15_Simple Shenker_Stanford_14_BHs_and_butterfly Shenker_Stanford_14_Mult... |
Entropic uncertainty relations and OTOCs occupy disparate subfields, but both quantify operator disagreement. We unite these quantifications, proving entropic uncertainty relations for QI scrambling (Theorem ). These relations make precise the extent to which scrambling drives operators away from compatibility. We then... |
Quasiprobabilities resemble probabilities but can behave nonclassically, assuming negative and nonreal values. The OTOC has been shown to equal an average over a quasiprobability distribution. Recent studies have uncovered several theoretical and experimental applications of the OTOC quasiprobability NYH_17_Jarzynski N... |
Our entropic uncertainty relations generalize in two ways. First, they extend from the famous four-point OTOC to arbitrarily high-order, arbitrarily out-of-time-ordered correlators. Such OTOCs reflect later, subtler stages of scrambling and equilibration Roberts_16_Chaos Haehl_17_Classification Haehl_17_Thermal NYH_18_... |
The rest of this paper is organized as follows. We briefly overview entropic uncertainty relations and OTOCs in Section In Sec. II , we reason intuitively to the form that entropic uncertainty relations for scrambling should assume. This intuition is made rigorous in Sec. III Numerical simulations of a spin chain illus... |
in Sec. IV We generalize to higher-point OTOCs in Sec. then to weak values beyond scrambling in Sec. VI Background Here, we review the two measures of operator disagreement, entropic uncertainty relations and OTOCs. |
I A Entropic uncertainty relations Heisenberg captured the complementarity of position and momentum in Heisenberg_27_Uber Kennard concretized this complementarity in the first uncertainty relation Kennard_27_Zur Robertson proved the uncertainty relation featured in many textbooks Robertson_29_Uncertainty |
[EQUATION] We have set [MATH] to one. [MATH] and [MATH] denote observables defined on a Hilbert space [MATH] The expectation value [MATH] is evaluated on a state [MATH] The standard deviation [MATH] |
quantifies the spread in the possible outcomes of a measurement of [MATH] The standard deviations have provoked objections (e.g., Deutsch_83_Uncertainty ). For example, consider relabeling the eigenvalues |
[MATH] of [MATH] Relabeling should not change the operators’ compatibility, but [MATH] can skyrocket. Stripping the [MATH] ’s off of [MATH] leaves a function of probabilities: Denote by [MATH] the probability that a measurement of [MATH] yields [MATH] On probability distributions [MATH] are defined entropies. Entropies... |
The Maassen-Uffink relation exemplifies entropic uncertainty relations Maassen_88_Generalized [EQUATION] The Shannon entropy is defined as |
[MATH] The maximum overlap [MATH] is defined in terms of the eigendecompositions [EQUATION] as [EQUATION] Hence the bound ( ) is independent of the eigenvalues [MATH] , as desired. The bound is tight if [MATH] is small. |
[MATH] is smallest when the eigenbases are mutually unbiased [MATH] wherein [MATH] denotes the Hilbert space’s dimensionality. For example, the Pauli operators |
[MATH] and [MATH] have mutually unbiased eigenbases. If you prepare any eigenstate of [MATH] then measure [MATH] you have no idea which outcome will obtain. Hence [MATH] and [MATH] are said to fail maximally to commute. Entropic uncertainty relations have applications to many topics in quantum theory, including quantum... |
I B Out-of-time-ordered correlators OTOCs reflect chaos and QI spreading in quantum many-body systems. Settings range from ultracold atoms and trapped ions to holographic black holes (e.g., LarkinO_69 Lashkari_13_Towards Kitaev_15_Simple Shenker_Stanford_14_BHs_and_butterfly Shenker_Stanford_14_Multiple_shocks Roberts_... |
[MATH] denotes the space of density operators, or trace-one positive-semidefinite linear operators, defined on [MATH] The OTOC has the form |
[EQUATION] for unitary and/or Hermitian [MATH] and [MATH] localized far apart. The OTOC forms the nontrivial component of [EQUATION] |
This magnitude-squared commutator equals [MATH] if [MATH] and [MATH] are unitary (e.g., Pauli operators). Several pieces of evidence imply that the OTOC signals chaos. We review a semiclassical argument about the butterfly effect: Classical chaos hinges on sensitivity to initial perturbations. Consider initializing a c... |
with a strong kick. Let the pendulum begin another trial at a nearby point [MATH] The pendulum follows different phase-space trajectories in the two trials. The trajectories diverge exponentially, as quantified with a Lyapunov exponent. |
The OTOC captures a similar divergence. Let us construct two protocols that differ largely by an initial perturbation. The system could consist of an [MATH] -site chain of spin- [MATH] |
degrees of freedom, or qubits Suppose that [MATH] is pure. Protocol I consists of (i) preparing the system in [MATH] (ii) perturbing the system with a local [MATH] |
(as by flipping spin 1 with [MATH] ), (iii) evolving the system under a nonintegrable Hamiltonian, (iv) perturbing with a local [MATH] |
(such as the final spin’s [MATH] ), and (v) evolving the system backward, under [MATH] This protocol prepares [MATH] Following protocol II, one prepares [MATH] |
and skips the initial [MATH] The system evolves forward under [MATH] is perturbed with [MATH] and reverse-evolves under [MATH] Only afterward does [MATH] perturb the system. Protocol II prepares [MATH] |
How much does the initial [MATH] perturbation affect the system’s final state? The answer manifests in the overlap [EQUATION] Nonlocal systems, such as the Sachdev-Ye-Kitaev (SYK) model Sachdev_93_Gapless Kitaev_15_Simple Polchinski_16_Spectrum Maldacena_16_Remarks obey the final relation. [In local systems, [MATH] dec... |
controls the exponential decay. Hence [MATH] reflects a Lyapunov-type divergence reminiscent of classical-chaotic sensitivity to initial perturbations. |
Smallness of [MATH] tends to reflect highly nonlocal entanglement. After [MATH] , no local probe [MATH] can recover information about any earlier, initially local perturbation [MATH] This many-body nonlocality is called scrambling |
Brown_13_Scrambling Hosur_16_Chaos II Intuitive construction of entropic uncertainty relations for quantum information scrambling |
Uncertainty relations and OTOCs, reflecting quantum operator disagreement in different subfields, cry out for unification. But how can one form an uncertainty relation for scrambling? One might try substituting [MATH] and [MATH] |
into the uncertainty relation ( ). But the bound would bear no signature of scrambling. Moreover, simulations imply, simple choices of [MATH] and [MATH] eigenbases fail to become mutually unbiased after [MATH] |
NYH_18_Quasi A clue suggests how entropic uncertainty relations for scrambling may be realized: The entropic inequality ( ) replaced the textbook inequality ( ). Inequality ( ) contains one commutator. The OTOC appears in a commutator’s squared magnitude [Eqs. ( ) and ( )]. Hence “squaring” Ineq. ( ), in some sense, mi... |
How might this “squaring” manifest? In the left-hand side (LHS) of Ineq. ( ), each entropy [MATH] depends on one operator, [MATH] or [MATH] Imagine “doubling” each operator by replacing it with two operators. The two operators suited to scrambling are [MATH] and [MATH] We therefore envision an entropy |
[MATH] defined in terms of a measurement of [MATH] followed by a measurement of [MATH] This replacement for [MATH] must differ from the replacement for [MATH] but the OTOC contains only two local operators. We therefore reverse the measurements: |
[MATH] The reversal mirrors the OTOC’s semiclassical interpretation, Eq. ( ). How can the right-hand side (RHS) of Ineq. ( be “squared”? |
[MATH] equals a product of two inner products. “Squaring” [MATH] creates a product of four inner products, or the trace of four outer products |
[MATH] Outer products generalize to projectors [MATH] Hence a trace of a product of four projectors, [MATH] should appear in an entropic uncertainty bound for scrambling. Such a trace is known to characterize scrambling. It forms the quasiprobability behind the OTOC |
NYH_17_Jarzynski NYH_18_Quasi Alonso_18_Out Quasiprobability distributions represent quantum states as probability distributions represent classical statistical-mechanical states. Like probabilities, quasiprobabilities are normalized to one. Yet quasiprobabilities violate axioms of probability theory, such as nonnegati... |
The OTOC equals an average over a quasiprobability distribution defined as follows NYH_17_Jarzynski NYH_18_Quasi The OTOC operators eigendecompose as |
[EQUATION] In the spin-chain example, the eigenvalues [MATH] The projector [MATH] projects onto the eigenvalue- [MATH] eigenspace of [MATH] |
[MATH] is defined analogously. Consider substituting from Eqs. ( into the OTOC definition ( ). Factoring out the sums and the eigenvalues yields |
[EQUATION] The OTOC equals an average over the OTOC quasiprobability, [EQUATION] The quasiprobability forms a distribution [MATH] This set of numbers contains more information than the OTOC, which follows from coarse-graining. |
[MATH] concretizes the relationship between scrambling and nonequilibrium statistical mechanics NYH_17_Jarzynski informs schemes for measuring the OTOC experimentally NYH_17_Jarzynski NYH_18_Quasi Dressel_18_Strengthening distinguishes scrambling from decoherence in measurements of open-system OTOCs Alonso_18_Out and u... |
[MATH] governs terms in the entropic uncertainty bound for scrambling. The quasiprobability tightens the bound when the system scrambles. We evaluate the quasiprobability on the identity operator [MATH] |
because entropic uncertainty bounds cannot depend on any state [MATH] Uncertainty relations require, moreover, that eigenvalues be “stripped off” of operators. |
[MATH] follows from stripping the eigenvalues off the OTOC, by Eq. ( ). We can predict the form of the uncertainty-bound term that will contain [MATH] Quasiprobabilities can be measured via weak measurement: An interaction Hamiltonian couples a detector to the system. A small coupling constant [MATH] governs the intera... |
[MATH] is extracted from the data through a high-order term. [MATH] should therefore appear in a high-order-in- [MATH] term in our entropic uncertainty bound. |
The uncertainty relation’s RHS contains [MATH] only if the LHS involves weak measurements. Consider measuring [MATH] weakly, then [MATH] strongly. Each possible pair [MATH] of outcomes has some probability of obtaining. On this probability, we propose to define the entropy |
[MATH] [MATH] should be defined similarly. Let us summarize our intuitive reasoning. Entropic uncertainty relations for scrambling should have the form |
[EQUATION] The exponent [MATH] [MATH] quantifies the uncertainty about the outcomes that follow from preparing an arbitrary [MATH] measuring [MATH] weakly, and then measuring [MATH] strongly. |
[MATH] results from reversing the measurement protocol. Having constructed expectations via intuition, we now prove them. III Formalization of entropic uncertainty relations for quantum information scrambling |
We introduce the setup in Sec. III A and formalize our measurements in Sec. III B The measurements yields outcomes distributed according to probability distributions on which are defined entropies in Sec. III C Our main result (Theorem is presented in Sec. III D |
and analyzed in Sec. III E III A Setup We continue to focus on a quantum many-body system illustrated with a chain of [MATH] qubits. To simplify notation, we omit hats from operators. Many-body quantities are defined as in the introduction: the Hilbert space [MATH] its dimensionality [MATH] the arbitrary state [MATH] t... |
(illustrated with [MATH] and [MATH] ), the Heisenberg-picture [MATH] the projectors [MATH] and [MATH] the eigenvalues [MATH] and [MATH] the OTOC [MATH] and the OTOC quasiprobability [MATH] |
The Hilbert space [MATH] is assumed to be discrete, in accordance with Tomamichel_12_Framework Krishna_01_Entropic whose results we use. Continuous-variable systems are addressed in Sec. VII We emphasize nonintegrable, nonlocal Hamiltonians. We assume that [MATH] and [MATH] are Hermitian, for simplicity, but the result... |
can be Hermitian and/or unitary NYH_17_Jarzynski NYH_18_Quasi If [MATH] is unitary but not Hermitian, for example, measurements of [MATH] are replaced with measurements of the Hermitian generator of [MATH] |
III B Formalization of measurements A sequence of [MATH] and [MATH] measurements forms a generalized measurement. Generalized measurements are formalized, in QI theory, with positive operator-valued measures (POVMs) NielsenC10 A POVM [MATH] consists of positive operators |
[MATH] that obey the completeness condition [MATH] [MATH] labels the outcomes. POVMs replace measurements of observables [MATH] and [MATH] |
in generalized entropic uncertainty relations Krishna_01_Entropic Tomamichel_12_Framework We adapt the formalism used by Tomamichel Tomamichel_12_Framework for concreteness and for ease of comparison with a standard reference. In Tomamichel_12_Framework appear POVMs illustrated with measurements of observables. |
These general POVMs manifest, in the context of scrambling, as follows. We label as “the forward measurement” a weak measurement of [MATH] followed by a projective measurement of [MATH] We use the term “weak measurement of [MATH] as in NYH_18_Quasi A projector [MATH] is effectively measured weakly. One can effectively ... |
by, e.g., coupling the detector to [MATH] and calibrating the detector appropriately. The experimenter chooses the value of [MATH] the choice directs the calibration. See Sec. IV A |
and NYH_18_Quasi, , Sec. I D 4) for example implementations. The reverse process constitutes the second POVM, for a definition of “reverse” that we concretize after formalizing the weak measurement. |
To measure [MATH] weakly, one prepares a detector in a state [MATH] The system’s [MATH] is coupled weakly to a detector observable, via an interaction unitary [MATH] A detector observable is measured projectively, yielding an outcome [MATH] |
The weak measurement induces dynamics modeled with Kraus operators NielsenC10 Preskill_15_Ch3 Kraus operators represent the system-of-interest evolution effected by a coupling to an ancilla, which effectively measures the system: |
[EQUATION] The operators satisfy the completeness relation [MATH] Let [MATH] temporarily denote the system’s precoupling state. The detector has a probability |
[MATH] of registering the outcome [MATH] The outcome-dependent [MATH] quantifies the interaction strength. The experimenter can tune [MATH] whose smallness reflects the measurement’s weakness: |
[MATH] We refer to various constants [MATH] as [MATH] ’s. Imagine strongly measuring the detector observable without having coupled the detector to the system. The outcome [MATH] has a probability [MATH] |
of obtaining. We invoke Kraus operators’ unitary equivalence Preskill_15_Ch3 to ensure that [MATH] The forward POVM [MATH] is defined through the composite Kraus operators |
[EQUATION] Recall that [MATH] projects onto the [MATH] eigenspace of [MATH] Each POVM element has the form [MATH] The reverse POVM, [MATH] is defined through the composite Kraus operators |
[EQUATION] To round out the reversal, we not only swap the [MATH] measurement with the [MATH] but also Hermitian-conjugate. Conjugation negates imaginary numbers. It represents, e.g., the time-reversal of magnetic fields. |
Let us clarify which variables are chosen and which vary randomly. [MATH] is a random outcome whose value varies from realization to realization of the forward POVM. |
[MATH] is a random outcome whose value varies from realization to realization of the reverse POVM. The experimentalist chooses the values of [MATH] and [MATH] Though a forward trial’s [MATH] and [MATH] |
can differ from a reverse trial’s [MATH] and [MATH] both protocols’ measurements [of [MATH] and of [MATH] are essentially the same. |
III C Entropies Consider preparing the system in the state [MATH] then measuring the forward POVM, [MATH] One prepares a detector in some fiducial state. Some detector observable is effectively coupled to the system’s [MATH] Then, some detector observable couples to a classical |
register. The register records an outcome [MATH] . Next, the system’s [MATH] couples to another classical register. This register records the outcome [MATH] |
The two-register system ends in the state [EQUATION] The eigenvalues, [MATH] form a probability distribution over the possible pairs [MATH] |
of measurement outcomes. Entropies of the distribution equal entropies of [MATH] The order- [MATH] Rényi entropy of a quantum state [MATH] is |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.