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arXiv:2108.00087 · Physical Review A

Fluctuation and dissipation in memoryless open quantum evolutions

Fabricio Toscano, G. M. Bosyk, Steeve Zozor, M. Portesi

Published Dec 22, 2021• 2 citations
arXivHTMLPDF

Abstract

The von Neumann entropy rate for open quantum systems is, in general, written in terms of entropy production and entropy flow rates, encompassing the second law of thermodynamics. For memoryless open quantum systems, whose evolutions correspond to one-parameter quantum dynamical semigroups, we find a decomposition of the infinitesimal generator of the dynamics that unveils the fluctuation-dissipation structure of these evolutions. Thus, the fluctuation part of this decomposition allows to relate the von Neumann entropy rate with the divergence-based quantum Fisher information, at any time. Applied to continuous-in-time quantum Gaussian channels, our decomposition leads to the quantum analog of the generalized classical de Bruijn identity, thus expressing the quantum fluctuation-dissipation relation in that kind of channels. Finally, from this perspective, we analyze how stationarity arises.

DOI
10.1103/physreva.104.062207

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The von Neumann entropy rate for open quantum systems is, in general, written in terms of entropy production and entropy flow rates, encompassing the second law of thermodynamics. For memoryless open quantum systems, whose evolutions correspond to one-parameter quantum dynamical semigroups, we find a decomposition of the infinitesimal generator of the dynamics that unveils the fluctuation-dissipation structure of these evolutions. Thus, the fluctuation part of this decomposition allows to relate the von Neumann entropy rate with the divergence-based quantum Fisher information, at any time. Applied to continuous-in-time quantum Gaussian channels, our decomposition leads to the quantum analog of the generalized classical de Bruijn identity, thus expressing the quantum fluctuation-dissipation relation in that kind of channels. Finally, from this perspective, we analyze how stationarity arises.
A quantum statistical expression for the entropy of a nonequilibrium system is defined so as to be consistent with Gibbs’ relation, and is shown to corresponds to dynamical variable by introducing analogous to the Heisenberg picture in quantum mechanics. The general relation between system-reservoir interactions and an entropy change operator in an open quantum system, relying just on the framework of statistical mechanics and the definition of von Neumann entropy. By using this formula, we can obtain the correct entropy production in the linear response framework. The present derivation of entropy production is directly based on the first principle of microscopic time-evolution, while the previous standard argument is due to the thermodynamic energy balance.
The Alicki-Lindblad-Fannes dynamical (ALF) entropy measures the rate at which new information is gathered about a quantum system by inspecting its long-time evolution. We propose an extension of the ALF entropy to open quantum dynamics as a measure of back-flow of information from the environment. Such a proposal is stronger than the existing ones based only on the open system reduced dynamics. In the case of a qubit collisionally coupled to a classical spin chain, we obtain an exact expression for the open-system ALF entropy\textit{open-system ALF entropy}open-system ALF entropy explicitly depending on the environment correlations. An extreme case shows how the information flow from environment to system corresponds to vanishing entropy production as for reversible finite quantum systems.
Thermodynamic force and irreversible current are the foundational concepts of classical nonequilibrium thermodynamics. Entropy production rate is provided by their product in classical systems, ranging from mesoscopic to macroscopic systems. However, there is no complete quantum extension of such a structure that respects quantum mechanics. In this paper, we propose anti-Hermitian operators that represent currents and forces accompanied by a gradient structure in open quantum systems described by the quantum master equation. We prove that the entropy production rate is given by the product of the force and current operators, which extends the canonical expression of the entropy production rate in the classical systems. The framework constitutes a comprehensive analogy with the nonequilibrium thermodynamics of discrete classical systems. We also show that the structure leads to the extensions of some results in stochastic thermodynamics: the geometric housekeeping-excess decomposition of entropy production and thermodynamic trade-off relations such as the thermodynamic uncertainty relation and the dissipation-time uncertainty relation. In discussing the trade-off relations, we will introduce a measure of fluctuation, which we term the quantum diffusivity.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
The irreversible motion of an open quantum system can be represented through an ensemble of state vectors following a stochastic dynamics with piecewise deterministic paths. It is shown that this representation leads to a natural definition of the rate of quantum entropy production. The entropy production rate is expressed in terms of the von Neumann entropy and of the numbers of quantum jumps corresponding to the various decay channels of the open system. The proof of the positivity and of the convexity of the entropy production rate is given. Monte Carlo simulations of the stochastic dynamics of a driven qubit and of a $\ensuremath{\Lambda}$ configuration involving a dark state are performed in order to illustrate the general theory.
We derive a universal inequality that provides a lower bound on the ensemble-averaged von Neumann entropy change in a quantum system subject to continuous measurement and dissipation. Our result clarifies how entropy production is fundamentally constrained by three distinct contributions: (i) the non-Hermitian structure of the dissipation operator, (ii) the standard variance associated with measurement-induced fluctuations, and (iii) a generalized quantum variance reflecting the noncommutativity between the measurement observable and the quantum state. This third term vanishes when the state and observable commute, and thus represents a purely quantum contribution arising from coherence disturbance and measurement backaction. The derived inequality generalizes classical information-thermodynamic relations, such as the Sagawa--Ueda inequality, to the quantum regime, providing a new perspective on the trade-offs between information acquisition, control, and entropy production in continuously monitored open quantum systems.