A family of monotone quantum relative entropies

Mathieu Lewin (1) , Julien Sabin (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We study here the elementary properties of the relative entropy $\cH(A,B)=\tr[\phi(A)-\phi(B)-\phi'(B)(A-B)]$ for $\phi$ a convex function and $A,B$ bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if $\phi'$ is operator monotone. We use this to appropriately define $\cH(A,B)$ in infinite dimension.

Domaines: Physique mathématique [math-ph]; Analyse fonctionnelle [math.FA]

Mathieu Lewin, Julien Sabin. A family of monotone quantum relative entropies. Letters in Mathematical Physics, 2014, 104 (6), pp.691-705. ⟨10.1007/s11005-014-0689-y⟩. ⟨hal-00862464⟩