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arXiv:2106.09870 · Physical review. E

Thermodynamic uncertainty relation for quantum first-passage processes

Yoshihiko Hasegawa

Published Apr 20, 2022• 19 citations
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Abstract

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.

DOI
10.1103/physreve.105.044127

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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 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.
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.
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.
The thermodynamic and kinetic uncertainty relations indicate trade-offs between the relative fluctuation of observables and thermodynamic quantities such as dissipation and dynamical activity. Although these relations have been well studied for classical systems, they remain largely unexplored in the quantum regime. In this Letter, we investigate such trade-off relations for Markovian open quantum systems whose underlying dynamics are quantum jumps, such as thermal processes and quantum measurement processes. Specifically, we derive finite-time lower bounds on the relative fluctuation of both dynamical observables and their first passage times for arbitrary initial states. The bounds imply that the precision of observables is constrained not only by thermodynamic quantities but also by quantum coherence. We find that the product of the relative fluctuation and entropy production or dynamical activity is enhanced by quantum coherence in a generic class of dissipative processes of systems with nondegenerate energy levels. Our findings provide insights into the survival of the classical uncertainty relations in quantum cases.
Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter θθθ and its corresponding extensive quantum operator O^\hat{O}O^ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter θθθ is quantified by its quantum Fisher information Fθ\mathcal{F}_θFθ​. The framework is based on an exact integral representation that relates Fθ\mathcal{F}_θFθ​ to the autocorrelation function of operator O^\hat{O}O^. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: Δθ ΔO‾≥kBTΔθ\,\overline{ΔO} \ge k_\text{B}TΔθΔO≥kB​T with ΔO‾≡∂θ⟨O^⟩ Δθ\overline{ΔO}\equiv\partial_θ\langle\hat{O}\rangle\,ΔθΔO≡∂θ​⟨O^⟩Δθ and TTT is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.
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.