Quantum Hellinger distances revisited
Abstract
This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:17771804, 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 +c B A σ B ), where σ is an arbitrary KuboAndo mean, and c ∈(0 ,1 ) is the weight of σ . We note that these divergences belong to the family of maximal quantum fdivergences, 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 1/2power 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.
 Publication:

Letters in Mathematical Physics
 Pub Date:
 March 2020
 DOI:
 10.1007/s11005020012820
 arXiv:
 arXiv:1903.10455
 Bibcode:
 2020LMaPh.110.2039P
 Keywords:

 Quantum Hellinger distance;
 KuboAndo mean;
 Weighted multivariate mean;
 Barycenter;
 Data processing inequality;
 Convexity;
 Mathematical Physics;
 Mathematics  Functional Analysis;
 Quantum Physics;
 47A64 (Primary);
 15A24;
 81Q10 (Secondary)
 EPrint:
 v2: Section 4 on the commutative case, and Subsection 5.2 on a possible measure of noncommutativity added, as well as references to the maximal quantum $f$divergence literature