Convexity of a certain operator trace functional

• URL: http://arxiv.org/abs/2109.11528v1
• Date: Thu, 23 Sep 2021 17:51:46 GMT
• Title: Convexity of a certain operator trace functional
• Authors: Eric Evert, Scott McCullough, Tea \v{S}trekelj, Anna Vershynina
• Abstract summary: In this article the operator trace function $ Lambda_r,s(A)[K, M] := operatornametr(K*Ar M Ar 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.
• Score: 1.1470070927586014
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $s\geq 1/2.$

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