Variational Representations related to Quantum R\'{e}nyi Relative Entropies

Abstract

In this paper, we focus on variational representations of some matrix symmetric norm functions that are related to the quantum R\'{e}nyi relative entropy. Concretely, we obtain variational representations of the function (A,B)\mapsto \normmm{(B^{q/2}K^*A^pKB^{q/2})^s} for symmetric norms by using the H\"{o}lder inequality and Young inequality. These variational expressions enable us to make the proofs of the convexity/concavity of the trace function (A,B)\mapsto \tr (B^{q/2}K^*A^pKB^{q/2})^s more clear.