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arXiv:2210.07866 · Journal of Statistical Physics

Stochastic Entropy Production: Fluctuation Relation and Irreversibility Mitigation in Non-unital Quantum Dynamics

Eliana Fiorelli, Stefano Gherardini, Stefano Marcantoni

Published Jun 14, 2023• 4 citations
arXivHTMLPDF

Abstract

In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in Phys Rev E 92:032129 (2015), we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in Phys Rev Res 2:033250 (2020), is analyzed in this context.

DOI
10.1007/s10955-023-03118-2

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In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in Phys Rev E 92:032129 (2015), we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in Phys Rev Res 2:033250 (2020), is analyzed in this context.
Abstract One of the major goals of quantum thermodynamics is the characterization of irreversibility and its consequences in quantum processes. Here, we discuss how entropy production provides a quantification of the irreversibility in open quantum systems through the quantum fluctuation theorem. We start by introducing a two-time quantum measurement scheme, in which the dynamical evolution between the measurements is described by a completely positive, trace-preserving (CPTP) quantum map (forward process). By inverting the measurement scheme and applying the time-reversed version of the quantum map, we can study how this backward process differs from the forward one. When the CPTP map is unital, we show that the stochastic quantum entropy production is a function only of the probabilities to get the initial measurement outcomes in correspondence of the forward and backward processes. For bipartite open quantum systems we also prove that the mean value of the stochastic quantum entropy production is sub-additive with respect to the bipartition (except for product states). Hence, we find a method to detect correlations between the subsystems. Our main result is the proposal of an efficient protocol to determine and reconstruct the characteristic functions of the stochastic entropy production for each subsystem. This procedure enables to reconstruct even others thermodynamical quantities, such as the work distribution of the composite system and the corresponding internal energy. Efficiency and possible extensions of the protocol are also discussed. Finally, we show how our findings might be experimentally tested by exploiting the state of-the-art trapped-ion platforms.
Abstract Thermodynamic irreversibility is well characterized by the entropy production arising from non-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a term that only depends on population unbalances, and one that is underpinned by quantum coherences. This allows us to identify a genuine quantum contribution to the entropy production in non-equilibrium quantum processes. We discuss how these features emerge both in Lindblad-Davies differential maps and finite maps subject to the constraints of thermal operations. We also show how this separation naturally leads to two independent entropic conservation laws for the global system-environment dynamics, one referring to the redistribution of populations between system and environment and the other describing how the coherence initially present in the system is distributed into local coherences in the environment and non-local coherences in the system-environment state. Finally, we discuss how the processing of quantum coherences and the incompatibility of non-commuting measurements leads to fundamental limitations in the description of quantum trajectories and fluctuation theorems.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.
Exploiting previous results on Markovian dynamics and fluctuation theorems, we study the consequences of memory effects on single realizations of nonequilibrium processes within an open system approach. The entropy production along single trajectories for forward and backward processes is obtained with the help of a recently proposed classical-like non-Markovian stochastic unravelling, which is demonstrated to lead to a correction of the standard entropic fluctuation theorem. This correction is interpreted as resulting from the interplay between the information extracted from the system through measurements and the flow of information from the environment to the open system: Due to memory effects, single realizations of a dynamical process are no longer independent and their correlations fundamentally affect the behavior of entropy fluctuations.
Abstract We investigate the total stochastic entropy production of a two-level bosonic open quantum system under protocols of time dependent coupling to a harmonic environment. These processes are intended to represent the measurement of a system observable, and consequent selection of an eigenstate, whilst the system is also subjected to thermalising environmental noise. The entropy production depends on the evolution of the system variables and their probability density function, and is expressed through system and environmental contributions. The continuous stochastic dynamics of the open system is based on the Markovian approximation to the exact, noise-averaged stochastic Liouville-von Neumann equation, unravelled through the addition of stochastic environmental disturbance mimicking a measuring device. Under the thermalising influence of time independent coupling to the environment, the mean rate of entropy production vanishes asymptotically, indicating equilibrium. In contrast, a positive mean production of entropy as the system responds to time dependent coupling characterises the irreversibility of quantum measurement, and a comparison of its production for two coupling protocols, representing connection to and disconnection from the external measuring device, satisfies a detailed fluctuation theorem.