Some Operator and Trace Function Convexity Theorems

TL;DR

This paper investigates convexity and concavity properties of trace functions involving positive matrices, settling many open questions and proving cases of the Audenaert-Datta Rényi entropy conjectures.

Contribution

It provides a comprehensive analysis of convexity and concavity of specific trace functions, resolving open problems and advancing understanding in matrix analysis and quantum information theory.

Findings

Concavity of trace functions is fully characterized.

Many convexity cases are settled, partially resolving open questions.

Certain cases of the Audenaert-Datta Rényi entropy conjectures are proved.

Abstract

We consider convex trace functions $\Phi_{p,q,s} = Trace[ (A^{q/2}B^p A^{q/2})^s]$ where $A$ and $B$ are positive $n\times n$ matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of $A^{q/2}B^p A^{q/2}$ and convexity/concavity of the closely related trace functional $Trace[ A^{q/2}B^p A^{q/2} C^r]$. For concavity, these questions are completely settled, thereby settling cases left open by Hiai, while the convexity questions are settled in many cases. As a consequence, the Audenaert-Datta R\'enyi entropy conjectures are proved for some cases.