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arXiv:2401.09936 · Physical review. E

Entropy production from maximum entropy principle: A unifying approach

Adalberto D. Varizi, Pedro Silva Correia

Published Aug 5, 2024• 2 citations
arXivHTMLPDF

Abstract

Entropy production is the crucial quantity characterizing irreversible phenomena and the second law of thermodynamics. Yet, a ubiquitous definition eludes consensus. Given that entropy production arises from incomplete access to information, in this work we use Jaynes' maximum entropy principle to establish a framework that brings together prominent and apparently conflicting definitions. More generally, our definition of entropy production addresses any tomographically incomplete quantum measurement and/or the action of a quantum channel on a system.

DOI
10.1103/physreve.110.024109

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Entropy production is the crucial quantity characterizing irreversible phenomena and the second law of thermodynamics. Yet, a ubiquitous definition eludes consensus. Given that entropy production arises from incomplete access to information, in this work we use Jaynes' maximum entropy principle to establish a framework that brings together prominent and apparently conflicting definitions. More generally, our definition of entropy production addresses any tomographically incomplete quantum measurement and/or the action of a quantum channel on a system.
Abstract The entropy production rate is a key quantity in nonequilibrium thermodynamics of both classical and quantum processes. No universal theory of entropy production is available to date, which hinders progress toward its full grasping. By using a phase space-based approach, here we take the current framework for the assessment of thermodynamic irreversibility all the way to quantum regimes by characterizing entropy production—and its rate—resulting from the continuous monitoring of a Gaussian system. This allows us to formulate a sharpened second law of thermodynamics that accounts for the measurement back action and information gain from a continuously monitored system. We illustrate our framework in a series of physically relevant examples.
The second law of classical equilibrium thermodynamics, based on the positivity of entropy production, asserts that any process occurs only in a direction that some information may be lost (flow out of the system) due to the irreversibility inside the system. However, any thermodynamic system can exhibit fluctuations in which negative entropy production may be observed. In particular, in stochastic quantum processes due to quantum correlations and also memory effects we may see the reversal energy flow (heat flow from the cold system to the hot system) and the backflow of information into the system that leads to the negativity of the entropy production which is an apparent violation of the Second Law. In order to resolve this apparent violation, we will try to properly extend the Second Law to quantum processes by incorporating information explicitly into the Second Law. We will also provide a thermodynamic operational meaning for the flow and backflow of information. Finally, it is shown that negative and positive entropy production can be described by a quantum thermodynamic force.
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics.This tutorial focuses on the derivation of the first and second law for isolated and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies.The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat, and temperature.These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes.The consistency with previous results is demonstrated.The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically.An integral fluctuation theorem for entropy production is satisfied.In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond.
The entropy shows an unavoidable tendency of disorder in thermostatistics according to the second thermodynamics law. This provides a minimization entropy principle for quantum thermostatistics with the von Neumann entropy and nonextensive quantum thermostatistics with special Tsallis entropy. Our goal in this work is to provide operational characterizations of general entropy measures. We present the first unified principle consistent with the second thermodynamics law in terms of general quantum entropies for both quantum thermostatistics and nonextensive quantum thermostatistics. This further reveals new features beyond the second thermodynamics law by maximization the cross entropy during irreversible measurement procedures. The present result is useful for asymptotical tasks of quantum entropy estimations and universal quantum source encoding without the state tomography. It is further applied to single-shot state transitions and cooling in quantum thermodynamics with limited information. These results should be interesting in the many-body theory and long-range quantum information processing.
Abstract The maximum entropy principle, as applied to quantum systems, is a fundamental prescript positing that for a quantum system for which we only have partial knowledge, the maximum entropy state consistent with the partial knowledge is a valuable choice as the system’s state. An intriguing result is that in case the only prior knowledge is of a fixed energy, the maximum entropy state turns out to be the thermal state, a ubiquitous state in several arenas, especially in statistical mechanics. We extend the consequences of this principle from quantum states to quantum processes. We establish that a quantum channel attains maximal output entropy under a fixed energy constraint if and only if it is an absolutely thermalizing channel, where the fixed output is the thermal state corresponding to that energy. Our results have potential implications for understanding the informational and thermodynamic utility of quantum channels under physical constraints. As an application, we examine the consequences for private randomness distillation from fixed energy constrained quantum processes.
In finite-dimensional quantum systems, temperature cannot be uniquely defined. This, in turn, implies that there are several ways to define entropy production in finite-dimensional quantum systems, because the classical entropy production depends on temperature. We propose a unified definition of entropy production based on the difference in quantum relative entropy with respect to reference states characterized by effective temperatures. We demonstrate that the proposed definition naturally decomposes into a Clausius-type entropy production and an additional contribution arising from the time dependence of the effective temperature. Furthermore, we show that requiring the entropy production rate to take the conventional form as the sum of the entropy change and the heat flow constrains the effective temperature to be either constant or equal to a specific energy-matching effective temperature. For general initial states, entropy production can become negative, in which case we derive lower bounds on entropy production and establish sufficient conditions for its non-negativity using the trace distance.