Convexity of a certain operator trace functional

TL;DR

This paper investigates the convexity and concavity properties of a specific operator trace functional, revealing conditions under which these properties hold or are lost, with implications for quantum information theory.

Contribution

The paper introduces the operator trace function $ ext{Lambda}_{r,s}$ and characterizes its convexity and concavity properties, especially under perturbations, connecting it to well-known operator functions.

Findings

$ ext{Lambda}_{r,s}$ is never concave.

Convexity holds only when $r=1$ and $s extgreater=1/2$.

Convexity is lost under small perturbations of matrices.

Abstract

In this article the operator trace function $ \Lambda_{r,s}(A)[K, M] := {\operatorname{tr}}(K^*A^r M A^r K)^s$ is introduced and its convexity and concavity properties are investigated. This function has a direct connection to several well-studied operator trace functions that appear in quantum information theory, in particular when studying data processing inequalities of various relative entropies. In the paper the interplay between $\Lambda_ {r,s}$ and the well-known operator functions $\Gamma_{p,s}$ and $\Psi_{p,q,s}$ is used to study the stability of their convexity (concavity) properties. This interplay may be used to ensure that $\Lambda_{r,s}$ is convex (concave) in certain parameter ranges when $M=I$ or $K=I.$ However, our main result shows that convexity (concavity) is surprisingly lost when perturbing those matrices even a little. To complement the main theorem, the convexity (concavity) domain of $\Lambda$ itself is examined. The final result states that $\Lambda_{r,s}$ is never concave and it is convex if and only if $r=1$ and $s\geq 1/2.$