On a new definition of quantum entropy

TL;DR

This paper introduces a new quantum entropy operator whose expectation value increases during non-adiabatic processes, thereby providing a quantum mechanical proof of the second law of thermodynamics and illustrating it with an exactly solvable example.

Contribution

It defines a quantum entropy operator that aligns with the second law, proving entropy increase during non-adiabatic transformations within quantum mechanics.

Findings

Quantum entropy expectation increases during non-adiabatic processes.

The second law is established as a quantum theorem.

The driven harmonic oscillator exemplifies the principle.

Abstract

It is proved here that, as a consequence of the unitary quantum evolution, the expectation value of a properly defined quantum entropy operator (as opposed to the non-evolving von Neumann entropy) can only increase during non adiabatic transformations and remains constant during adiabatic ones. Thus Clausius formulation of the second law is established as a theorem in quantum mechanics, in a way that is equivalent to the previously established formulation in terms of minimal work principle [A. E. Allahverdyan and T. M. Nieuwenhuizen, Phys. Rev. E 71, 046107 (2005)]. The corresponding Quantum Mechanical Principle of Entropy Increase is then illustrated with an exactly solvable example, namely the driven harmonic oscillator. Attention is paid to both microcanonical and canonical initial condition. The results are compared to their classical counterparts.