A family of monotone quantum relative entropies

Abstract

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