# Quantum Hellinger distances revisited

## TL;DR

This paper introduces generalized quantum Hellinger divergences involving Kubo-Ando means, explores their properties, and characterizes barycenters, clarifying previous claims about their form in non-commuting cases.

## Contribution

It extends quantum Hellinger distances by defining a family of divergences with Kubo-Ando means and characterizes their barycenters, correcting prior assumptions for non-commuting operators.

## Key findings

- Generalized divergences are jointly convex and satisfy data processing inequality.
- Barycenters are characterized as weighted multivariate 1/2-power means in commuting cases.
- The previously claimed barycenter form does not hold for non-commuting operators.

## Abstract

This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences, that are of the form $\phi(A,B)=\mathrm{Tr} \left((1-c)A + c B - A \sigma B \right),$ where $\sigma$ is an arbitrary Kubo-Ando mean, and $c \in (0,1)$ is the weight of $\sigma.$ 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 (DPI). 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 the work of Bhatia et al. mentioned above, is true in the case of commuting operators, but it is not correct in the general case.