Speed Limit for Classical Stochastic Processes

Shiraishi, Naoto; Funo, Ken; Saito, Keiji

doi:10.1103/physrevlett.121.070601

Cited by 221 publications

(220 citation statements)

References 45 publications

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“…This relation may suggest further connection between finite-time quantum control theory and QSLs. ( ( ) ( )) ( ) L p p A 0 , 2 2 with ò s S ≔ṫ d and t t å -( ( ) ( )) ≔ | ( ) ( )| L p p p p 0 , 0 n n n , which was reported in [19].…”

Section: Resultsmentioning

confidence: 84%

“…For an isolated quantum system, the velocity term can be related to the energy fluctuation as is the case for the Mandelstam-Tamm type QSL [2]. In [19], we have related the velocity term to the quantities that appear in stochastic thermodynamics and derived the speed limit for classical stochastic processes. However, for open quantum systems, most of the QSLs derived in the literatures remain as a formal mathematical expression, and physical intuition is hard to extract from the obtained results.…”

Section: Introductionmentioning

confidence: 99%

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Speed limit for open quantum systems

2019

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We study the quantum speed limit for open quantum systems described by the Lindblad master equation. The obtained inequality shows a trade-off relation between the operation time and the physical quantities such as the energy fluctuation and the entropy production. We further identify a quantity characterizing the speed of the state transformation, which appears only when we consider the open system dynamics in the quantum regime. When the thermal relaxation is dominant compared to the unitary dynamics of the system, we show that this quantity is approximated by the energy fluctuation of the counter-diabatic Hamiltonian which is used as a control field in the shortcuts to adiabaticity protocol. We discuss the physical meaning of the obtained quantum speed limit and try to give better intuition about the speed in open quantum systems. t .

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Section: Resultsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Speed limit for open quantum systems

2019

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“…. This means that a small determinant can be achieved if the transition rates 9 While preparing this manuscript we became aware of independent result by Shiraishi et al [30], which implies proposition 2; see appendix B for details. 10 For two probability distributions p, q over n states, the 1 ℓ norm is defined p q…”

Section: Limits Of Ctmcsmentioning

confidence: 95%

“…However, this does not affect any result of our paper, and the proposition remains valid exactly as written. This is because the proposition can be derived from the result equation (14) of [1] ( [30] in the original text). In fact, the steps between equation (14) of [1] and proposition 2 of our paper were already described in our published appendix, but only as an aside immediately following our (mistaken) 'proof' (very bottom of page 14).…”

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confidence: 99%

Number of hidden states needed to physically implement a given conditional distribution

2019

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4 5 Massachusetts Institute of Technology. ( ) . The result follows by combining equations (1) and (2) and taking p=p(0), Pp=p(τ). ORCID iDsJeremy A Owen https:/ /orcid.org/0000-0002-9180-3794 Artemy Kolchinsky https:/ /orcid.org/0000-0002-3518-9208 References[1] Shiraishi N, Funo K and Saito K 2018 Speed limit for classical stochastic processes Phys. Rev. Lett. 121 070601 AbstractWe consider the problem of how to construct a physical process over a finite state space X that applies some desired conditional distribution P to initial states to produce final states. This problem arises often in the thermodynamics of computation and nonequilibrium statistical physics more generally (e.g. when designing processes to implement some desired computation, feedback controller, or Maxwell demon). It was previously known that some conditional distributions cannot be implemented using any master equation that involves just the states in X. However, here we show that any conditional distribution P can in fact be implemented-if additional 'hidden' states not in X are available. Moreover, we show that it is always possible to implement P in a thermodynamically reversible manner. We then investigate a novel cost of the physical resources needed to implement a given distribution P: the minimal number of hidden states needed to do so. We calculate this cost exactly for the special case where P represents a single-valued function, and provide an upper bound for the general case, in terms of the non-negative rank of P. These results show that having access to one extra binary degree of freedom, thus doubling the total number of states, is sufficient to implement any P with a master equation in a thermodynamically reversible way, if there are no constraints on the allowed form of the master equation. (Such constraints can greatly increase the minimal needed number of hidden states.) Our results also imply that for certain P that can be implemented without hidden states, having hidden states permits an implementation that generates less heat.where p(0) and p(1) are column vectors whose entries sum to one, representing probability distributions at times t=0 and t=1 respectively. If our system has n states, the matrix P is an n×n stochastic matrix, representing the conditional distribution of the system state at the final time given its state at the initial time. The entry P ij is the probability that the system is in state i at the final time given that it was in state j at the initial time.

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“…This is because the proposition can be derived from the result equation (14) of [1] ([30] in the original text). In fact, the steps between equation (14) of [1] and proposition 2 of our paper were already described in our published appendix, but only as an aside immediately following our (mistaken) 'proof' (very bottom of page 14). For clarity, we reproduce those steps below.…”

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confidence: 97%