Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance

TL;DR

This paper introduces a new family of quantum fidelities based on Riemannian geometry, generalizing existing measures and exploring their properties, invariances, and extensions to divergences.

Contribution

It presents a Riemannian geometric framework for quantum fidelities, unifying and extending several known fidelity measures and divergences with new theoretical insights.

Findings

Generalized fidelity encompasses standard quantum fidelities.

Proves invariance and covariance properties of the new fidelity.

Extends the framework to multivariate and Rènyi divergence settings.

Abstract

We introduce a family of fidelities, termed generalized fidelity, which are based on the Riemannian geometry of the Bures-Wasserstein manifold. We show that this family of fidelities generalizes standard quantum fidelities such as Uhlmann-, Holevo-, and Matsumoto-fidelity and demonstrate that it satisfies analogous celebrated properties. The generalized fidelity naturally arises from a generalized Bures distance, the natural distance obtained by linearizing the Bures-Wasserstein manifold. We prove various invariance and covariance properties of generalized fidelity as the point of linearization moves along geodesic-related paths. We also provide a Block-matrix characterization and prove an Uhlmann-like theorem, as well as provide further extensions to the multivariate setting and to quantum R\'enyi divergences, generalizing Petz-, Sandwich-, Reverse sandwich-, and Geometric-R\'enyi divergences of order $\alpha$.