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arXiv:1808.08305

Bound on the rate of entropy change in open quantum systems

Pharnam Bakhshinezhad, A. T. Rezakhani

Published Aug 24, 2018• 1 citation
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Abstract

We study the temporal rate of variations of the von Neumann entropy in an open quantum system which interacts with a bath. We show that for almost all initial states of the bath and the system, the time-average of the rate of entropy change is bounded by a function which depends on various properties of the system and environment, and is mostly relatively small. This result holds true under fairly general conditions in almost any arbitrary quantum system.

DOI
10.48550/arxiv.1808.08305

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We study the temporal rate of variations of the von Neumann entropy in an open quantum system which interacts with a bath. We show that for almost all initial states of the bath and the system, the time-average of the rate of entropy change is bounded by a function which depends on various properties of the system and environment, and is mostly relatively small. This result holds true under fairly general conditions in almost any arbitrary quantum system.
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Control of open quantum dynamics is of great interest for realizing quantum technologies. Therefore, it is an important task to quantify and characterize the entropy for open quantum systems under decoherence. In this paper, we study the time evolution of the von Neumann entropy for open quantum systems described by the Lindblad master equation. Note that, in particular, when the decoherence corresponds to the measurement for the observable in the system, the von Neumann entropy tends to monotonically increases as the variance becomes larger. Furthermore, we present a lower bound of the von Neumann entropy in the long-time limit. This lower bound has advantages of being straightforwardly calculated and applicable to a general Markovian open quantum system.
The dynamics near a hyperbolic fixed point in phase space is modeled by an inverted harmonic oscillator. We investigate the effect of the classical instability on the open quantum dynamics of the oscillator, introduced through the interaction with a thermal bath, using both the survival probability function and the rate of von Neumann entropy increase, for large times. In this parameter range we prove, using influence functional techniques, that the survival probability function decreases exponentially at a rate ${\ensuremath{\kappa}}^{\ensuremath{'}}$ depending not only on the measure of instability in the model but also on the strength of interaction with the environment. We also show that ${\ensuremath{\kappa}}^{\ensuremath{'}}$ determines the rate of the von Neumann entropy increase and that this result is independent of the temperature of the environment. This generalizes earlier results that are valid in the limit of vanishing dissipation. The validity of inferring similar rates of survival probability decrease and entropy increase for quantum chaotic systems is also discussed.
Exact results about the nonequilibrium thermodynamics of open quantum systems at arbitrary timescales are obtained by considering all possible variations of initial conditions of a system. First we obtain a quantum-information theoretic equality for entropy production, valid for an arbitrary initial joint state of system and environment. For any finite-time process with a fixed initial environment, we then show that the system's loss of distinction-relative to the minimally dissipative state-exactly quantifies its thermodynamic dissipation. The quantum component of this dissipation is the change in coherence relative to the minimally dissipative state. Implications for quantum state preparation and local control are explored. For nonunitary processes-like the preparation of any particular quantum state-we find that mismatched expectations lead to divergent dissipation as the actual initial state becomes orthogonal to the anticipated one.
Thermalization in open systems coupled to macroscopic environments is usually analyzed from the perspective of relaxation of the reduced state of the system to the equilibrium state. Less emphasis is given to the change of the state of the bath. However, as previously shown for some specific models, during the thermalization the environment may undergo a nontrivial dynamics, indicated by the the change of its von Neumann entropy, at timescales even longer than the relaxation time of the system; here such a behavior is called post-thermalization. We provide a more detailed analysis of this phenomenon by simulating the full dynamics of a variety of systems together with their environment. In particular, the post-thermalization is qualitatively explained as a result of reconversion of the initially built up correlation between the system and the bath into the correlation between the degrees of freedom in the environment. We also present exemplary systems in which such a reconversion is suppressed due to non-Markovian dynamics or the presence of interactions.
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.