Bures-Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subset

Understand

In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space.

• He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures.
• In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds.
• The analog of the Hellinger distance turns out to be the Bures-Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory.