Statistical entropy of quantum systems

TL;DR

This paper explores the conditions under which von Neumann entropy accurately reflects thermodynamic entropy in quantum systems, addressing foundational debates and providing numerical analysis of a spin-1/2 system.

Contribution

It clarifies the assumptions needed for von Neumann entropy to represent thermodynamic entropy and revisits criticisms, offering new insights into quantum statistical mechanics.

Findings

Von Neumann entropy's equivalence to thermodynamic entropy depends on subtle assumptions.

Addressed criticisms of von Neumann entropy like time-invariance and subadditivity.

Numerical analysis of a spin-1/2 system supports theoretical arguments.

Abstract

Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.