  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.
