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Bures–Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subset
Citation Link: https://doi.org/10.15480/882.4758
Publikationstyp
Journal Article
Date Issued
2022-09-22
Sprache
English
Author(s)
Institut
TORE-DOI
Journal
Volume
5
Issue
2
Start Page
405
End Page
425
Citation
Information Geometry 5 (2): 405-425 (2022)
Publisher DOI
Scopus ID
Publisher
Springer Singapore
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.
Subjects
Bures distance
Information geometry
Optimal transport
Positive-definite matrices
Quantum information
Wasserstein metric
DDC Class
004: Informatik
600: Technik
Publication version
publishedVersion
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