10.1007/s11005-020-01282-0
Pitrik, Jozsef
Jozsef
Pitrik
Virosztek, Daniel
Daniel
Virosztek
Quantum Hellinger distances revisited
Springer Nature
2020
2020-03-25T15:57:48Z
2020-07-27T12:36:44Z
journal_article
https://research-explorer.app.ist.ac.at/record/7618
https://research-explorer.app.ist.ac.at/record/7618.json
0377-9017
1903.10455
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 ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of 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.