Concavity of certain matrix trace and norm functions

TL;DR

This paper advances the understanding of matrix trace and norm functions by establishing their concavity and convexity properties using refined mathematical techniques, with implications for quantum information theory.

Contribution

It introduces refined methods to prove joint concavity/convexity of matrix functions and extends these properties to symmetric norms involving operator means.

Findings

Proved joint concavity/convexity of extended Lieb type trace functions.

Established convexity properties of symmetric (anti-) norm functions with operator means.

Improved convexity results for norm functions using Carlen and Lieb's variational method.

Abstract

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2}}^s$, where $\Phi$ and $\Psi$ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form $||{\Phi(A^p)\sigma\Psi(B^q)}^s||$ involving an operator mean $\sigma$. Carlen and Lieb's variational method is also used to improve the convexity property of norm functions $||\Phi(A^p)^s||$.