On reduction of the wave-packet, decoherence, irreversibility and the second law of thermodynamics

TL;DR

This paper proves a quantum version of the second law of thermodynamics, demonstrating that quantum Boltzmann entropy increases under decoherence, supported by models including wave-packet reduction and quantum chaotic systems.

Contribution

It introduces a rigorous proof of quantum entropy increase due to decoherence and analyzes its behavior in various quantum models, extending the understanding of irreversibility.

Findings

Quantum Boltzmann entropy increases with decoherence.

Monotonic and non-monotonic entropy behaviors are demonstrated.

Decoherence enforces macroscopic purity in quantum chaotic systems.

Abstract

We prove a quantum version of the second law of thermodynamics: the (quantum) Boltzmann entropy increases if the initial (zero time) density matrix decoheres, a condition generally satisfied in Nature. It is illustrated by a model of wave-packet reduction, the Coleman-Hepp model, along the framework introduced by Sewell in his approach to the quantum measurement problem. Further models illustrate the monotonic-versus-non-monotonic behavior of the quantum Boltzmann entropy in time. As a last closely related topic, decoherence, which was shown by Narnhofer and Thirring to enforce macroscopic purity in the case of quantum K systems, is analysed within a different class of quantum chaotic systems, viz. the quantum Anosov models as defined by Emch, Narnhofer, Sewell and Thirring. A review of the concept of quantum Boltzmann entropy, as well as of some of the rigorous approaches to the quantum measurement problem within the framework of Schr\"{o}dinger dynamics, is given, together with an overview of the C* algebra approach, which encompasses the relevant notions and definitions in a comprehensive way.