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arXiv · 2304.04361

Equality cases in monotonicity of quasi-entropies, Lieb's concavity and Ando's convexity

Fumio Hiai
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Abstract

We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let $Φ:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$ be a trace-preserving map such that $Φ^*$ is a Schwarz map. When $f$ is an operator monotone or operator convex function on $[0,\infty)$, we present several equivalent conditions for the equality $S_f^K(Φ(ρ)\|Φ(σ))=S_f^{Φ^*(K)}(ρ\|σ)$ to hold for given positive operators $ρ,σ$ on $\mathcal{H}$ and $K\in\mathcal{B}(\mathcal{K})$. The conditions include equality cases in the monotonicity versions of Lieb's concavity and Ando's convexity theorems. Specializing the map $Φ$ we have equivalent conditions for equality cases in Lieb's concavity and Ando's convexity. Similar equality conditions are discussed also for monotone metrics and $χ^2$-divergences. We further consider some types of linear preserver problems for those quantum information quantities.

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Fumio Hiai. 2023-07-04. Equality cases in monotonicity of quasi-entropies, Lieb's concavity and Ando's convexity. https://arxiv.org/abs/2304.04361

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