  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. First we present an existing derivation of the Riemannian
metric and geodesics associated with this distance. Subsequently, we present a
novel derivation of the Riemannian distance and geodesics for this metric on
the subset of trace-one matrices, analogous to the Fisher distance for
probability measures.
