  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.
