This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form $$\phi (A,B)=\mathrm {Tr} \left( (1-c)A + c B - A \sigma B \right) ,$$ ϕ ( A , B ) = Tr ( 1 - c ) A + c B - A σ B , where $$\sigma $$ σ is an arbitrary Kubo–Ando mean, and $$c \in (0,1)$$ c ∈ ( 0 , 1 ) is the weight of $$\sigma .$$ σ . We note that these divergences belong to the family of maximal quantum f -divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case.

中文翻译：

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重新审视量子海灵格距离

这篇简短的笔记旨在研究 Bhatia 等人最近研究的量子海灵格距离。(Lett Math Phys 109:1777–1804, 2019) 特别强调重心。我们介绍了形式为 $$\phi (A,B)=\mathrm {Tr} \left( (1-c)A + c B - A \sigma B \right) 的广义量子 Hellinger 散度族， $$ ϕ ( A , B ) = Tr ( 1 - c ) A + c B - A σ B ，其中 $$\sigma $$ σ 是任意 Kubo-Ando 均值，$$c \in (0,1 )$$ c ∈ ( 0 , 1 ) 是 $$\sigma .$$ σ 的权重。我们注意到这些散度属于最大量子 f 散度族，因此是联合凸的，并且满足数据处理不等式。我们推导出这些广义量子海灵格散度的有限多个正定算符的重心表征。我们注意到重心的表征为加权多元 1/2 幂平均值，这在 Bhatia 等人中提出。(2019)，在通勤运营商的情况下是正确的，但在一般情况下是不正确的。