# Axiomatic characterization of the quantum relative entropy and free energy

## TL;DR

This paper axiomatizes quantum relative entropy with full-rank second argument and shows that free energy uniquely measures athermality under certain quantum thermodynamic conditions.

## Contribution

It provides an axiomatic foundation for quantum relative entropy and establishes the uniqueness of free energy as a measure of athermality in quantum thermodynamics.

## Key findings

- Quantum relative entropy is characterized by four simple axioms.
- Free energy is uniquely determined as a measure of athermality.
- Free energy is monotonic under extended Gibbs-preserving maps.

## Abstract

Building upon work by Matsumoto, we show that the quantum relative entropy with full-rank second argument is determined by four simple axioms: i) Continuity in the first argument, ii) the validity of the data-processing inequality, iii) additivity under tensor products, and iv) super-additivity. This observation has immediate implications for quantum thermodynamics, which we discuss. Specifically, we demonstrate that, under reasonable restrictions, the free energy is singled out as a measure of athermality. In particular, we consider an extended class of Gibbs-preserving maps as free operations in a resource-theoretic framework, in which a catalyst is allowed to build up correlations with the system at hand. The free energy is the only extensive and continuous function that is monotonic under such free operations.