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Electronic Journal of Linear Algebra

Some notes on quantum Hellinger divergences with Heinz means

Nguyen Lam, Rebecca Milley

Published Nov 10, 2020• 2 citations
Publisher

Abstract

The information geometry, convexity, in-betweenness property and the barycenter problem of quantum Hellinger divergences with Heinz means is studied. The limiting cases are also considered.
DOI
10.13001/ela.2020.5461

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In this paper, we introduce new quantum divergences of the form Φ(A,B)=Tr(AσB−AτB),A,B>0,where σ and τ are Kubo–Ando operator means such that σ≥τ. More precisely, we show that Φ(A,B) is a quantum divergence when σ is the weighted Kubo–Ando matrix power mean and τ is the weighted geometric mean. In addition, we construct a new quantum Hellinger-type divergence using the linear approximation of the function x. We also study the least-squares problem, the data processing inequality, and in-betweenness property of the matrix means with respect to the quantum Hellinger-type divergence.