Oostrum, Jesse vanJesse vanOostrum2022-10-172022-10-172022-09-22Information Geometry 5 (2): 405-425 (2022)http://hdl.handle.net/11420/13783In 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.en2511-249X20222405425Springer Singaporehttps://creativecommons.org/licenses/by/4.0/Bures distanceInformation geometryOptimal transportPositive-definite matricesQuantum informationWasserstein metricInformatikTechnikJournal Article10.15480/882.475810.1007/s41884-022-00069-710.15480/882.4758Journal Article