text stringlengths 128 2.05k |
|---|
This broadening has two more important significances: one for OTOC theory and one for weak-measurement theory. First, the extension reconciles the OTOC’s [MATH] with the tiny perturbation that triggers violent consequences in the classical butterfly effect: |
[MATH] can naturally be regarded, our uncertainty relations show, as being measured weakly. The weak measurement is perturbative literally, in the coupling strength [MATH] |
Within measurement theory, second, we have uncovered a physical significance of weak values [MATH] and Kirkwood-Dirac quasiprobabilities: These quantities govern first-order terms in entropic uncertainty relations obeyed by weak measurements. Quantum information theory therefore sheds light on mathematical objects whos... |
In a recent paper, an uncertainty relation was extended to unitaries, then applied to bound the OTOC Bong_18_Strong OTOC bounds have been known to limit the speed at which many-body entanglement can develop Maldacena_15_Bound Sekino_08_Fast Lashkari_13_Towards The present work takes a fundamentally different approach: ... |
This work uncovers several research opportunities. Inspired by condensed matter, we have focused on discrete systems. Also continuous systems—quantum field theories (QFTs)—have OTOCs used to study, e.g., black holes in the anti-de-Sitter-space/conformal-field-theory (AdS/CFT) duality Kitaev_15_Simple Shenker_Stanford_1... |
Second, Theorems and can be tested experimentally. The techniques needed exist: OTOC measurements have been proposed in detail Swingle_16_Measuring Yao_16_Interferometric Zhu_16_Measurement Bohrdt_16_Scrambling Campisi_16_Thermodynamics Tsuji_17_Exact NYH_17_Jarzynski NYH_18_Quasi Tsuji_18_Out and early-stage OTOC-meas... |
should be feasible in the immediate future, especially through the weak-measurement proposal for inferring the OTOC quasiprobability [MATH] |
NYH_17_Jarzynski NYH_18_Quasi Prospective platforms include superconducting qubits, ultracold atoms, trapped ions, quantum dots, and potentially NMR. |
Testing Theorem experimentally requires even fewer resources: Interacting many-body systems are unnecessary, and one weak measurement per trial suffices. Tantalizingly, though, two Piacentini_16_Measuring Suzuki_16_Observation Thekkadath_16_Direct |
and three Chen_18_Experimental sequential weak measurements have been realized recently. They can be applied to (i) characterize higher-order terms in Eqs. ( 52 and ( 53 ), (ii) test entropic uncertainty relations for higher-point OTOCs (Sec. ), and (iii) test entropic uncertainty relations for POVMs of sequential weak... |
Third, the entropic uncertainty relations for scrambling can be smoothed with an error tolerance [MATH] When smoothing, one ignores highly unlikely events Renner_05_Security Highly unlikely outcomes of weak-measurement experiments correspond to anomalous weak values and nonclassical quasiprobability values Dressel_15_W... |
[MATH] might actually tighten the spin-chain bound ( 25 ). Like smoothing, conditioning generalizes the entropic uncertainty relations in Tomamichel_12_Framework Consider holding a memory [MATH] that is entangled with a to-be-measured state [MATH] Conditioning on [MATH] can change your uncertainty about the measurement... |
in conditioned entropic uncertainty relations for scrambling. Finally, nonclassicality of [MATH] and [MATH] might strengthen the uncertainty bounds. The quasiprobability behaves nonclassically by acquiring negative real and nonzero imaginary components. The weak value [MATH] behaves nonclassically by lying outside the ... |
NYH_18_Quasi, , Sec. III and Sec. V A) Hence [MATH] cannot influence the bound. Second, [MATH] assumes negative values, but not when [MATH] Higher-point-OTOC quasiprobabilities could avoid this roadblock, and assume negative values in the bound, as higher-point forward and reverse protocols depend on weak [MATH] measur... |
Acknowledgements. We are grateful for conversations with Fernando G. S. L. Brandão, Sean Carroll, Justin Dressel, Patrick Hayden, José Raúl Gonzalez Alonso, Renato Renner, Brian Swingle, and Marco Tomamichel. NYH is grateful for support from the Institute for Quantum Information and Matter (IQIM), for a Barbara Groce G... |
Appendix A Proof of Theorem Tomamichel presents an entropic uncertainty relation for the smooth entropies [MATH] and [MATH] Tomamichel_12_Framework, , Result 7) Krishna and Parthasarathy derive one for |
[MATH] and [MATH] Krishna_01_Entropic, , Corollary 2.6) and Rastegin presents one for [MATH] and [MATH] Rastegin_08_Uncertainty, , Ineq. (13)) |
(proved in Rastegin_08_Statement ). We use Tomamichel’s notation, for concreteness. But the three uncertainty relations have the same RHSs. Hence our use of Tomamichel_12_Framework, , Result 7) |
translates directly into uses of the other two bounds. Tomamichel considers POVMs [MATH] and [MATH] whose outcomes are recorded in classical registers [MATH] and [MATH] The systems [MATH] and [MATH] can hold quantum side information about, or have correlations with, [MATH] and [MATH] An agent performing an information-... |
[EQUATION] in Tomamichel_12_Framework (see also Tomamichel_12_Tight Thinh_12_Tomographic Berta_16_Smooth ). The smooth entropies [MATH] and [MATH] |
follow from extremizing [MATH] and [MATH] The POVM overlap [MATH] , defined in Eq. ( A3 ), generalizes the overlap ( ). We set the smoothing parameter [MATH] to zero. We also trivialize the conditioning, setting the states of [MATH] Let us substitute in the POVMs ( 13 ) and ( 14 ): |
[EQUATION] The POVM overlap [MATH] is defined as [EQUATION] The operator norm has the form [EQUATION] The outer square-root equals, by Eqs. ( 13 ) and ( 14 ), |
[EQUATION] The two central projectors have collapsed into one: [MATH] The operator [MATH] is Hermitian and so eigendecomposes. The eigenvalues are real and nonnegative, being the squares of the singular values of |
[MATH] Also a physical argument implies the eigenvalues’ reality and nonnegativity: [MATH] is proportional to a quantum state: [MATH] |
represents the state that is maximally mixed over the eigenvalue- [MATH] eigenspace of [MATH] Imagine preparing [MATH] subjecting the state to the quantum channel defined by the operation elements NielsenC10 |
[MATH] subjecting the state to the channel defined by [MATH] and then measuring [MATH] projectively. The resultant state, [MATH] , is proportional to [MATH] The proportionality factor equals [MATH] the joint probability that (i) this realization of the initial channel’s action is labeled by [MATH] (ii) this realization... |
[MATH] is positive semidefinite and [MATH] equals a probability, the eigenvalues of [MATH] are real and nonnegative. The eigenvectors of [MATH] are eigenvectors of [MATH] |
[MATH] has two distinct eigenvalues [MATH] [MATH] , of degeneracy [MATH] and [MATH] , of degeneracy [MATH] Let [MATH] denote the [MATH] |
[MATH] eigenvalue associated with any eigenvector in the [MATH] eigenspace of [MATH] If [MATH] denotes the degeneracy of [MATH] [MATH] (We have omitted the [MATH] dependence from the symbol [MATH] for notational simplicity.) Every eigenvalue-0 eigenvector of [MATH] is an eigenvalue-0 eigenvector of [MATH] |
[MATH] Hence [MATH] eigendecomposes as [EQUATION] We use this eigenvalue decomposition to evaluate the RHS of Eq. ( A4 ), working from inside to outside. The outer square-root has the form |
[MATH] The projectors project onto orthogonal subspaces, so [MATH] We take the trace, [MATH] then exponentiate: [MATH] The limit as [MATH] gives the RHS of Eq. ( A4 ): |
[EQUATION] Only the greatest eigenvalue to survives: [MATH] But [MATH] is neither a parameter chosen by the experimentalist nor obviously experimentally measurable. Hence bounding the entropies with [MATH] |
is useless. Probabilities and quasiprobabilities are measurable. [MATH] equals a combination of probabilities and quasiprobabilities. We therefore seek to shift the [MATH] of Eq. ( A8 inside the [MATH] and the [MATH] Equivalently, we seek to shift the [MATH] of Eq. ( A9 inside the [MATH] We do so at the cost of introdu... |
[EQUATION] for all [MATH] This inequality follows from the Schatten [MATH] -norm’s monotonicity. The Schatten [MATH] -norm of an operator [MATH] is defined as |
[MATH] for [MATH] As [MATH] increases, the Schatten norm decreases monotonically: [EQUATION] Let [MATH] and [MATH] Raising each side of Ineq. ( A11 to the [MATH] power yields Ineq. ( A10 ). Applying Ineq. ( A10 ) to Eq. ( A9 bounds the operator norm as |
[EQUATION] We have invoked the trace’s cyclicality and [MATH] Substituting into Eq. ( A3 ) bounds the overlap: [EQUATION] We substitute into the trace from Eqs. ( 13 ) and ( 14 ): |
[EQUATION] Multiplying out yields [EQUATION] Six of the traces are instances of [MATH] Only the first term is constant in [MATH] If [MATH] is small, therefore, the maximum obtains where the first term maximizes, where [MATH] Hence every RHS term is implicitly evaluated at [MATH] |
We take the log of each side of Ineq. ( A15 ). The log’s monotonicity implies [MATH] We negate each side, then shift the negative sign across the max (as negative logs evoke entropies): |
[MATH] With this inequality and with Ineq. ( A15 ), we bound the RHS of Ineq. ( A1 ). Next, we factor out the [MATH] and invoke the log law for multiplication: |
[MATH] We then Taylor-approximate in the [MATH] ’s. The quasiprobability values are assumed to be small enough not to undermine the Taylor approximation. This assumption is reasonable: OTOC quasiprobability values [MATH] have not been observed in any of the numerical simulations performed for this paper or for NYH_18_Q... |
Appendix B Analytical calculations for the spin-chain example Let us derive the results in Sec. IV B We calculate the detector probability |
[MATH] the weak-measurement Kraus operators [MATH] the coupling strengths [MATH] and the entropies [MATH] Detector probability [MATH] |
Consider preparing the detector in [MATH] then measuring [MATH] The measurement has a probability [MATH] of yielding a position within [MATH] of [MATH] By Eq. ( 32 ), |
[EQUATION] Equation ( B1 ) determines the condition under which the uncertainty bound is nontrivial. The term [EQUATION] dominates the bound ( ). The trace equals [MATH] The bound is positive when |
[MATH] The min, acting on Eq. ( B1 ), chooses [MATH] We substitute in from Eq. ( B1 ), then solve for [MATH] [EQUATION] Inequality ( B3 ) does not violate Heisenberg’s measurement-disturbance uncertainty relation Heisenberg_27_Uber A finite time separates the [MATH] preparation from the [MATH] measurement. Yet the [MAT... |
[MATH] Bernien_17_Probing The rubidium atom has a mass of [MATH] kg. Denoting Boltzmann’s constant by [MATH] , we approximate [MATH] The momentum |
[MATH] stands in for [MATH] Lengths in periodic arrays can be measured with X-ray diffraction. Precisions of up to [MATH] have been achieved with silicon Mohr_12_CODATA Massa_11_Measurement (Though silicon lattices differ from rubidium arrays, both numbers reflect precision achievable with quantum experiments today.) S... |
[MATH] Approximately the same number of rubidium atoms formed the quantum simulator in Bernien_17_Probing Weak-measurement Kraus operators |
[MATH] and coupling strengths [MATH] The Kraus operators have the form (to within a global phase) [EQUATION] We redefine the Kraus operators such that the coefficient is real: |
[EQUATION] The outcome-dependent coupling is [EQUATION] We chose [MATH] , which (with [MATH] [MATH] , and [MATH] satisfies Ineq. ( B3 ). |
Appendix C Choice of [MATH] in the spin-chain example Equation (60) on p. 15 of NYH_18_Quasi motivates our choice. [MATH] appears, there, as a combination of correlators of [MATH] and [MATH] Let us set [MATH] and replace [MATH] with [MATH] We recall that [MATH] that the Pauli operators’ traces vanish, and that the Paul... |
[EQUATION] Let us analyze the expression piecemeal. First, the [MATH] at early times, because the influence from [MATH] has not reached [MATH] Random-matrix-theory cancellations suppress |
[MATH] at late times. Second, the OTOC begins at [MATH] and drops to [MATH] Third, suppose that [MATH] Combining these three behaviors, we infer the behavior of the RHS of Eq. ( C1 ). The first three terms sum to [MATH] The final term rises from [MATH] to [MATH] Therefore, [MATH] rises from [MATH] to [MATH] |
This rise strengthens the bound [MATH] [MATH] contributes to the bound through the term [EQUATION] in line ( 30 ). We have chosen for the couplings to have large imaginary parts, so |
[MATH] is dominated by [MATH] The quasiprobability is real for all arguments NYH_18_Quasi, , p. 24) Around [MATH] , therefore, ( C2 rises from [MATH] to |
[MATH] tightening the bound. In summary, the final [MATH] value in ( 30 points to [MATH] as a condition under which the uncertainty bound is relatively tight. We arbitrarily chose [MATH] |
Why should the first two [MATH] terms in ( 30 not guide our choice of [MATH] These terms influence the bound through [EQUATION] By Eq. ( C1 ), |
[MATH] As argued earlier, [MATH] is small at early and late times. Hence [MATH] for all [MATH] Hence ( C3 ) is expected to be negative, loosening the bound, regardless of our choices of [MATH] and [MATH] |
Appendix D Calculations: Qubit example for the weak-value uncertainty relation [MATH] can be weakly measured as follows. The detector is prepared in the state [MATH] [MATH] -controlled [MATH] conditions a rotation of the detector’s state on the system’s state. The interaction Hamiltonian |
[MATH] generates the unitary [EQUATION] The detector’s [MATH] is measured strongly, yielding the outcome [MATH] We can begin assembling the ingredients in Ineq. ( 52 ). The coupling-free probabilities [MATH] |
for [MATH] Next, we calculate the weak-measurement Kraus operators [MATH] and the outcome-dependent couplings [MATH] En route to [MATH] we define the physically equivalent |
[EQUATION] We remove a global phase: [EQUATION] To first order in [MATH] [EQUATION] The outcome-dependent coupling has the form [EQUATION] |
The weak value has the form [MATH] The nonreality is nonclassical Dressel_12_Significance Let us calculate the bound ( 53 ). [MATH] contains a factor |
[MATH] When this factor maximizes at [MATH] the minimum in [MATH] is attained. The probability [MATH] for all [MATH] Substituting into Eq. ( 53 ) yields |
[EQUATION] Having evaluated the RHS of Ineq. ( 52 ), we turn to the LHS. We calculate the POVM probabilities, then their entropies. |
[MATH] denotes an arbitrary system state, exemplified by [MATH] POVM II consists of a strong [MATH] measurement. The possible outcomes [MATH] have probabilities |
[MATH] of obtaining. If [MATH] , then [MATH] , and [MATH] The max entropy is [MATH] Smoothing cannot change this value. POVM I consists of a weak [MATH] measurement followed by a strong [MATH] measurement. The possible outcome tuples [MATH] |
correspond to the probabilities [MATH] We substitute in, then multiply out: [EQUATION] If [MATH] the distribution is uniform: [MATH] for all [MATH] Hence [MATH] Nor can smoothing alter this value. |
# Source: arxiv 1806.04223 # Title: Reversibility in space, time, and computation: the case of underwater acoustic communications # Sections: all # Downloaded: 2026-03-03T02:18:10.824083+00:00 |
Reversibility in space, time, and computation: the case of underwater acoustic communications Work in Progress Report Abstract Time reversal of waves has been successfully used in communications, sensing and imaging for decades. The application in underwater acoustic communications is of our special interest, as it put... |
Keywords: Acoustic time reversal Digital signal processing Lattice gas Reversible cellular automata Reversible circuits. Introduction |
The idea of wave time reversal has been considered for decades: among other references, we can find an early mention in Rolf Landauer’s work |
. The theory and practice of modern time reversal of waves stems from Mathias Fink’s idea of time reversal mirrors . While the first theoretical and practical results came from the case of sound waves (acoustics), the concept was translated in electromagnetic domain as well, through applications in optics |
and radio technology . The particular scenario we are considering here is the case of underwater acoustic communications (UAC), an application where the poor electromagnetic wave propagation makes sound waves the best solution. |
Fluid dynamics (motion of liquids and gases) using reversible cellular automata (RCA) has been discussed extensively in the past |
. However, this century has seen only a few applications of RCA to macro-scale engineering problems such as acoustic underwater sensing |
. Similarly, proofs of concept for hardware and software implementations of reversible digital signal processing exist , and we yet have to see it applied. This work will combine these underutilised methods of modelling and hardware implementation and relate them to time reversal, another physical form of reversibility... |
The style of exposition is tailored to give the first introduction to time reversal of waves to reversible computation community. Linking the two fields in both analytic (RCA) and synthetic sense (reversible hardware) is the main contribution of this paper: in future publications we will present the results. The wide a... |
State of the Art: Time Reversal The concept of (acoustic) time reversal is illustrated in Fig. . If we place a sound source in a heterogeneous medium within a cavity and let it emit a pulse, this pulse will travel through the medium and reach an array of transducers placed on the cavity walls (Fig. (a)). If we emit wha... |
Covering the whole cavity with transceivers is not feasible: not only does it ask for a large number of transceivers, but sometimes the system is deployed in (partially) open space, not a cavity. Hence, the option of a localised time reversal mirror (TRM) has to be considered: only a few transceivers co-located in a si... |
: the wave will pass through every point in space eventually and collect all the environment information on the way to the mirror. |
The application of this concept to communications is straightforward: the wave, when returned to the original sender may convey information from the receiver (TRM) and it will be focused only at the location of the original source (preventing both eavesdropping and interference at other locations). While there are othe... |
Modelling and Quantification Time reversal in the UAC setting is an example of a reversible process in a nominally reversible environment. While dynamics of water (or any fluid for that purpose) subject to sound waves, streams, waves and other motions are inherently reversible, most of the sources of the water dynamics... |
RCA give us such an option through the lattice gas models : cellular automata obeying the laws of fluid dynamics described by the Navier-Stokes equation. One such model, the celebrated FHP (Frisch-Hasslacher-Pomeau) lattice gas |
has had several improvements after its original statement in 1986 , but its basic form is simple and yet following the Navier-Stokes equations exactly. This is a model defined on a hexagonal grid through a set of rules of particle collision shown in Fig. . The model can be interpreted as an RCA via partitioning approac... |
, but the randomness of transitions when collisions include more possible outcomes (as seen in the figure) has to be taken into account. |
The FHP lattice gas provides us a two-dimensional model for UAC, easily implementable in software and capturing the necessary properties of the reversible medium. It is not a novel idea to use a lattice gas to model water, but neither acoustic underwater communications or time reversal of waves have been observed throu... |
The model we observe consists of the original source (transmitter) which causes the spread of an acoustic wave, the original sink (receiver) waiting for the wave to reach it, as well as scatterers and constant flows (streams) in the environment. The constant stream and the loss of information caused by some wave compon... |
The amplitude of the peak will fluctuate based on the location of the original source and may serve as a metric: a measure of reversibility. If we move the source over the whole surface of the model and measure this metric (whose analogue in quantum reversibility studies is fidelity or Loschmidt Echo |
) we obtain a heatmap of the surface with respect to the quality of time reversal. In the context of time reversal studies, it is used as a measure of the quality of communication, but in a more general context it can measure reversibility of a cellular automaton. The functionality of the model increases if we observe ... |
Reversible Hardware Implementation The reversibility in time of the communication scheme we use and the reversibility in space of the medium both suggest that the reversibility in computation should exist as well. Fig. gives an overview of a reversible architecture we are proposing, which consists of speakers/microphon... |
From the electromechanical point of view, a microphone and a speaker are the same device, running on the same physical principle, which makes the two ends of the scheme equivalent. The next element, the AD converter on one and DA converter on the other end are traditionally made in an irreversible fashion as the signal... |
The signal received is manipulated in the Fourier (frequency) domain by conjugation (change of the sign of the complex image’s phase) as conjugation in frequency domain results in time reversal in time domain. This asks for a chain of transform, manipulation and inverse transform so the new time domain signal can be em... |
. Hence, the FFT block can be considered reversible, and the Inverse Fast Fourier Transfom (IFFT) is just the FFT block with the reversed flow. Finally, the phase reversal is simply changing the sign of the half of the outputs coming from the FFT block, as the whole set of outputs comprises of phase and amplitude of th... |
Once we have determined the reversibility of the scheme, we note its symmetry as well. If we fold the structure in the middle (at the conjugation block), the same hardware can be used both to propagate the inputs and the outputs. While the particular details of circuit implementation are left for future work, where det... |
Conclusion In this work in progress report, we have presented the potential of reversible computation for time reversal in UAC. Future work will focus on both the modelling prospects using RCA and the reversible circuit implementation of the time reversal hardware. While going into more detail to cover all the practica... |
# Source: arxiv 1806.04256 # Title: Pseudorandom Generators for Width-3 Branching Programs # Sections: all # Downloaded: 2026-03-03T01:44:53.998308+00:00 |
Pseudorandom Generators for Width-3 Branching Programs Abstract We construct pseudorandom generators of seed length [MATH] that [MATH] -fool ordered read-once branching programs (ROBPs) of width [MATH] and length [MATH] For unordered ROBPs, we construct pseudorandom generators with seed length [MATH] This is the first ... |
Our constructions are based on the “iterated milder restrictions” approach of GMR 12 (which further extends the Ajtai-Wigderson framework AW85 ), combined with the INW-generator INW94 at the last step (as analyzed by BRRY14 ). For the unordered case we combine iterated milder restrictions with the generator of CHHL18 |
Two conceptual ideas that play an important role in our analysis are: 1. relabeling technique allowing us to analyze a relabeled version of the given branching program, which turns out to be much easier. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.