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Irreversibility, thermodynamic uncertainty relation, and information scrambling

Yoshihiko Hasegawa

Published Jan 18, 2021• 0 citations
OpenAlex

Abstract

Entropy production characterizes irreversibility. This viewpoint induces us to regard thermodynamic uncertainty relation, stating that higher precision can be achieved at the cost of higher entropy production, as a relation between precision and the extent of irreversibility. In this Letter, we show that the precision of arbitrary Hermitian observables has a lower bound comprising the Loschmidt echo, which represents the irreversibility of quantum dynamics. Considering the continuous measurement in quantum Markov processes, the obtained relation can be viewed as a lower bound for the precision of counting observables, which is a quantum analog of thermodynamic uncertainty relation. Moreover, applying the obtained relation to the information scrambling, we obtain an upper bound for the out-of-time-order correlator.

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Entropy production characterizes irreversibility. This viewpoint induces us to regard thermodynamic uncertainty relation, stating that higher precision can be achieved at the cost of higher entropy production, as a relation between precision and the extent of irreversibility. In this Letter, we show that the precision of arbitrary Hermitian observables has a lower bound comprising the Loschmidt echo, which represents the irreversibility of quantum dynamics. Considering the continuous measurement in quantum Markov processes, the obtained relation can be viewed as a lower bound for the precision of counting observables, which is a quantum analog of thermodynamic uncertainty relation. Moreover, applying the obtained relation to the information scrambling, we obtain an upper bound for the out-of-time-order correlator.
Entropy production characterizes irreversibility. This viewpoint allows us to consider the thermodynamic uncertainty relation, which states that a higher precision can be achieved at the cost of higher entropy production, as a relation between precision and irreversibility. Considering the original and perturbed dynamics, we show that the precision of an arbitrary counting observable in continuous measurement of quantum Markov processes is bounded from below by the Loschmidt echo between the two dynamics, representing the irreversibility of quantum dynamics. When considering particular perturbed dynamics, our relation leads to several thermodynamic uncertainty relations, indicating that our relation provides a unified perspective on classical and quantum thermodynamic uncertainty relations.
Symmetry breaking with respect to the time reversal characterizes entropy production. This viewpoint induces us to regard thermodynamic uncertainty relation (TUR), stating that higher precision can be achieved at the cost of higher entropy production, as a relation between precision and the extent of symmetry breaking. In this Letter, we formalize a TUR as symmetry breaking in quantum Markov processes, where the extent of symmetry breaking constitutes the thermodynamic cost. We derive a TUR for quantum continuous measurement, which holds for any conceivable continuous measurement. We demonstrate that counting- and current-type TURs can be obtained as particular cases of the obtained bound.
We derive a thermodynamic uncertainty relation for first passage processes in quantum Markov chains. We consider first passage processes stopping after a fixed number of jumps, which contrasts with typical quantum Markov chains which end at a fixed time. We bound observables of first passage processes in quantum Markov chains by the Loschmidt echo, which quantifies the extent of irreversibility in quantum many-body systems. Considering a particular case of the bound, we show that the lower bound corresponds to the quantum Fisher information, which has played a fundamental role in uncertainty relations in quantum systems. Moreover, our bound reduces to a thermodynamic uncertainty relation for classical first passage processes when considering classical dynamics.
We derive a thermodynamic uncertainty relation for first passage processes in quantum Markov chains. We consider first passage processes that stop after a fixed number of jump events, which contrasts with typical quantum Markov chains which end at a fixed time. We obtain bounds for the observables of the first passage processes in quantum Markov chains by the Loschmidt echo, which quantifies the extent of irreversibility in quantum many-body systems. Considering a particular case, we show that the lower bound corresponds to the quantum Fisher information, which plays a fundamental role in uncertainty relations in quantum systems. Moreover, considering classical dynamics, our bound reduces to a thermodynamic uncertainty relation for classical first passage processes.
We derive a thermodynamic uncertainty relation for general open quantum dynamics, described by a joint unitary evolution on a composite system comprising a system and an environment. By measuring the environmental state after the system-environment interaction, we bound the counting observables in the environment by the survival activity, which reduces to the dynamical activity in classical Markov processes. Remarkably, the relation derived herein holds for general open quantum systems with any counting observable and any initial state. Therefore, our relation is satisfied for classical Markov processes with arbitrary time-dependent transition rates and initial states. We apply our relation to continuous measurement and the quantum walk to find that the quantum nature of the system can enhance the precision. Moreover, we can make the lower bound arbitrarily small by employing appropriate continuous measurement.
For classic systems, the thermodynamic uncertainty relation (TUR) states that the fluctuations of a current have a lower bound in terms of the entropy production. Some TURs are rooted in information theory, particularly derived from relations between observations (mean and variance) and dissimilarities, such as the Kullback-Leibler divergence, which plays the role of entropy production in stochastic thermodynamics. We generalize this idea for quantum systems, where we find a lower bound for the uncertainty of quantum observables given in terms of the quantum relative entropy. We apply the result to obtain a quantum thermodynamic uncertainty relation in terms of the quantum entropy production, valid for arbitrary dynamics and nonthermal environments.