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

Quantum corrections to the entropy and its application in the study of quantum Carnot engines