Quantum coarse-grained entropy and thermalization in closed systems

TL;DR

This paper explores the properties of Observational entropy in quantum systems, demonstrating its potential to serve as a foundation for understanding thermodynamic behavior and the second law in quantum mechanics.

Contribution

It provides a comprehensive analysis, proofs, and simulations of quantum Observational entropy, highlighting its properties, interpretations, and relevance to thermodynamics.

Findings

Observational entropy can involve non-commuting coarse-grainings without losing properties.

It rises to the thermodynamic value in isolated quantum systems.

Supports a quantum foundation for the second thermodynamic law.

Abstract

We investigate the detailed properties of Observational entropy, introduced by \v{S}afr\'{a}nek et al. [Phys. Rev. A 99, 010101 (2019)] as a generalization of Boltzmann entropy to quantum mechanics. This quantity can involve multiple coarse-grainings, even those that do not commute with each other, without losing any of its properties. It is well-defined out of equilibrium, and for some coarse-grainings it generically rises to the correct thermodynamic value even in a genuinely isolated quantum system. The quantity contains several other entropy definitions as special cases, it has interesting information-theoretic interpretations, and mathematical properties -- such as extensivity and upper and lower bounds -- suitable for an entropy. Here we describe and provide proofs for many of its properties, discuss its interpretation and connection to other quantities, and provide numerous simulations and analytic arguments supporting the claims of its relationship to thermodynamic entropy. This quantity may thus provide a clear and well-defined foundation on which to build a satisfactory understanding of the second thermodynamical law in quantum mechanics.