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

TL;DR

This paper derives variational representations for matrix symmetric norm functions related to quantum Rényi relative entropy, aiding in understanding their convexity and concavity properties.

Contribution

It provides new variational formulas for matrix functions associated with quantum Rényi entropy using classical inequalities.

Findings

Derived variational representations for symmetric norm functions.

Clarified convexity/concavity proofs of related trace functions.

Connected variational formulas to quantum information measures.

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.