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arXiv:1612.01179 · Journal of Mathematical Physics

On the limit relation for the quantum relative entropy

József Zsolt Bernád, Béla A. Frigyik

Published Jun 1, 2017• 0 citations
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Abstract

Recently, Diósi et al. introduced a simple, yet very interesting model for reservoirs, in order to study the relationship between thermodynamic entropy production of a system and the corresponding von Neumann entropy (or informatic entropy, as it was called by the authors). They came up with a conjecture about the asymptotic behaviour of the state introduced in the model. The conjecture was proven later by Csiszár et al. In this paper, we give a different interpretation of the original question and extend the result to a collection of more general states.

DOI
10.1063/1.4986246

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Recently, Diósi et al. introduced a simple, yet very interesting model for reservoirs, in order to study the relationship between thermodynamic entropy production of a system and the corresponding von Neumann entropy (or informatic entropy, as it was called by the authors). They came up with a conjecture about the asymptotic behaviour of the state introduced in the model. The conjecture was proven later by Csiszár et al. In this paper, we give a different interpretation of the original question and extend the result to a collection of more general states.
In a thermodynamic model, Diósi et al. [Int. J. Quantum Inf. 4, 99–104 (2006)] arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the Belavkin-Staszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is introduced for classical-quantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture.
We consider the generic model of a finite-size quantum electron system connected to two (temperature and particle) reservoirs. The quantum open system is driven out of equilibrium by the presence of both potential temperature and chemical differences between the two reservoirs. The nonequilibrium (NE) thermodynamical properties of such a quantum open system are studied for the steady state regime. In such a regime, the corresponding NE density matrix is built on the so-called generalised Gibbs ensembles. From different expressions of the NE density matrix, we can identify the terms related to the entropy production in the system. We show, for a simple model, that the entropy production rate is always a positive quantity. Alternative expressions for the entropy production are also obtained from the Gibbs–von Neumann conventional formula and discussed in detail. Our results corroborate and expand earlier works found in the literature.
Traditional measures of entropy, like the Von Neumann entropy, while fundamental in quantum information theory, are insufficient when interpreted as thermodynamic entropy due to their invariance under unitary transformations, which contradicts observed entropy increases in isolated systems. Recognizing this limitations of existing measures for thermodynamic entropy, recent research has focused on observational entropy (OE) as a promising alternative, offering practical applicability and theoretical insights. In this work, we extend the scope of observational entropy by generalizing it to a parameterized version called ααα-Observational entropy (ααα-OE). ααα-OE is expressed in terms of the Petz-Rényi relative entropy between the states on which a quantum-to-classical channel is applied. The ααα-OE reduces to OE under α→1α\rightarrow 1α→1. We prove various properties of the ααα-OE, which are the generalization of the properties of OE, including the monotonically increasing of ααα-OE as a function of refinement of coarse-graining. We further explore the role of ααα-OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.