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arXiv:1806.11055

Dissipation production in a closed two-level quantum system as a test of the obversibility of the dynamics

Claudia L. Clarke, Ian J. Ford

Published Jun 28, 2018• 1 citation
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Abstract

Irreversible behaviour is traditionally associated with open stochastic dynamical systems, but an asymmetry in the probabilistic specification of a closed deterministic system can similarly lead to a disparity between the likelihoods of a particular forward and corresponding backward behaviour starting from a specified time. Such a comparison is a test of a property denoted obversibility, which may be quantified in terms of dissipation production as a measure of irreversibility. We here discuss the procedure needed to evaluate dissipation production in a simple, deterministic two-level quantum system described by a statistical ensemble of state vectors and then provide numerical results for illustrative situations. We consider cases that both do and do not fulfill an Evans-Searles Fluctuation Theorem for the dissipation production, and identify conditions for which the system will display time-asymmetric average behaviour as it evolves.

DOI
10.48550/arxiv.1806.11055

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Irreversible behaviour is traditionally associated with open stochastic dynamical systems, but an asymmetry in the probabilistic specification of a closed deterministic system can similarly lead to a disparity between the likelihoods of a particular forward and corresponding backward behaviour starting from a specified time. Such a comparison is a test of a property denoted obversibility, which may be quantified in terms of dissipation production as a measure of irreversibility. We here discuss the procedure needed to evaluate dissipation production in a simple, deterministic two-level quantum system described by a statistical ensemble of state vectors and then provide numerical results for illustrative situations. We consider cases that both do and do not fulfill an Evans-Searles Fluctuation Theorem for the dissipation production, and identify conditions for which the system will display time-asymmetric average behaviour as it evolves.
Irreversible behavior in open stochastic dynamical systems is quantified by stochastic entropy production, a property that measures the difference in likelihoods of forward and subsequent backward system evolution. But for a closed system, governed by deterministic dynamics, such an approach is not appropriate. Instead, we can consider the difference in likelihoods of forward and "obverse" behavior: the latter being a backward trajectory initiated at the same time as the forward trajectory. Such a comparison allows us to define "dissipation production," an analog of stochastic entropy production. It quantifies the breakage of a property of the evolution termed "obversibility" just as stochastic entropy production quantifies a breakage of reversibility. Both are manifestations of irreversibility. In this study we discuss dissipation production in a quantum system. We consider a simple, deterministic, two-level quantum system characterized by a statistical ensemble of state vectors, and we provide numerical results to illustrate the ideas. We consider cases that both do and do not satisfy an Evans-Searles Fluctuation Theorem for the dissipation production, and hence identify conditions under which the system displays time-asymmetric average behavior: an arrow of time.
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.
We investigate the entropic consequences of the relaxation of an open two-level quantum system towards a thermalised statistical state, using a framework of quantum state diffusion with evolution described by a minimal set of raising and lowering Lindblad operators. We demonstrate that thermalisation is typically accompanied by a persistent non-zero mean rate of change of the environmental component of stochastic entropy production. This thermodynamic signature can be associated with the purification of the reduced density matrix ρ of the randomly evolving state under these dynamics, which may be contrasted with the impurity of the more frequently considered ensemble average of ρ. The system adopts stationary statistics, with zero stochastic entropy production, after a further stage of relaxation once purity has been achieved. We show that apparent pathological mathematical difficulties in the computation of stochastic entropy production emerge in this evolution towards stationarity if ρ is represented using a certain set of coordinates, though these problems can be removed by choosing a different set. We conclude that frameworks for modelling open systems must be carefully selected to provide satisfactory thermodynamic as well as dynamic behaviour.
We investigate stochastic entropy production in a two-level quantum system that performs Rabi oscillations while undergoing quantum measurement brought about by continuous random disturbance by an external measuring device or environment. The dynamics produce quantum Zeno and anti-Zeno effects for certain measurement regimes, and the stochastic entropy production is a measure of the irreversibility of the behaviour. When the strength of the measurement disturbance is time-dependent, the stochastic entropy production separates into three components. Two represent relaxational behaviour, one being specific to systems represented by coordinates that are odd under time reversal symmetry, and a third characterises the nonequilibrium stationary state arising from breakage of detailed balance in the dynamics. The study illustrates how the ideas of stochastic thermodynamics may be applied in similar ways to both quantum and classical systems.
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.
The thermodynamic behavior of Markovian open quantum systems can be described at the level of fluctuations by using continuous monitoring approaches. However, practical applications require assessing imperfect detection schemes, where the definition of main thermodynamic quantities becomes subtle and universal fluctuation relations are unknown. Here, we fill this gap by deriving a universal fluctuation relation that links thermodynamic entropy production and information-theoretical irreversibility along single trajectories in inefficient monitoring setups. This relation provides as a corollary an irreversibility estimator of dissipation using imperfect detection records that lower bounds the underlying entropy production at the level of visible trajectories. We illustrate our findings with a driven-dissipative two-level system following quantum jump trajectories and discuss the experimental applicability of our results for thermodynamic inference.