Von Neumann's information engine without the spectral theorem

TL;DR

This paper reinterprets von Neumann's entropy derivation by replacing the spectral theorem with operational assumptions, extending the analysis to broader theories and clarifying the operational meaning of information gain.

Contribution

It shows that von Neumann's entropy formula can be derived without the spectral theorem, using only repeatability and reversibility, applicable to non-spectral theories.

Findings

Operational assumptions suffice for entropy derivation

Groenewold--Ozawa information gain as a monotone

Extension of entropy concepts beyond quantum theory

Abstract

Von Neumann obtained the formula for the entropy of a quantum state by assuming the validity of the second law of thermodynamics in a thought experiment involving semipermeable membranes and an ideal gas of quantum-labeled particles. Despite being operational in the most part, von Neumann's argument crucially departs from an operational narrative in its use of the spectral theorem. In this work we show that the role of the spectral theorem in von Neumann's argument can be taken over by the operational assumptions of repeatability and reversibility, and using these we are able to explore the consequences of the second law also in theories that do not possess a unique spectral decomposition. As a byproduct, we obtain the Groenewold--Ozawa information gain as a natural monotone for a suitable ordering of instruments, providing it with an operational interpretation valid in quantum theory and beyond.