On the relation between classical and quantum-mechanical entropy

Benjamin W. Schumacher, Michael D. Westmoreland

Akira Sone, Sebastian Deffner

The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.

Élliott H. Lieb, Mary Beth Ruskai

Akira Sone, Sebastian Deffner

The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. We prove that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state. This insight is exploited to define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both, quantum as well as classical scenarios, is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.

Tian Qiu, Zhaoyu Fei, Rui Pan +1 more

Entropy is one of the most basic concepts in thermodynamics and statistical mechanics. The most widely used definition of statistical mechanical entropy for a quantum system is introduced by von Neumann. While in classical systems, the statistical mechanical entropy is defined by Gibbs. The relation between these two definitions of entropy is still not fully explored. In this work, we study this problem by employing the phase-space formulation of quantum mechanics. For those quantum states having well-defined classical counterparts, we study the quantum-classical correspondence and quantum corrections of the entropy. We expand the von Neumann entropy in powers of ℏ by using the phase-space formulation, and the zeroth-order term reproduces the Gibbs entropy. We also obtain the explicit expression of the quantum corrections of the entropy. Moreover, we find that for the thermodynamic equilibrium state, all terms odd in ℏ are exactly zero. As an application, we derive quantum corrections for the net work extraction during a quantum Carnot cycle. Our results bring important insights into the understanding of quantum entropy and may have potential applications in the study of quantum heat engines.

Ugo Farinelli, A. Gamba