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

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 Rényi divergences, generalizing Petz-, Sandwich-, Reverse sandwich-, and Geometric-Rényi divergences of order $α$.

💡 Research Summary

The paper introduces a novel family of quantum fidelities, called “generalized fidelity,” which are derived from the Riemannian geometry of the Bures‑Wasserstein (BW) manifold. The authors start by reviewing classical and quantum fidelities—Uhlmann, Holevo, Matsumoto, log‑Euclidean, and the z‑family—and point out that each can be viewed as a particular way of pairing the arithmetic and geometric means of positive semidefinite matrices. They then observe that the BW manifold of positive‑definite matrices carries a natural Riemannian metric, whose geodesics, exponential and logarithmic maps, and distance function are all known in closed form.

The central construction is to linearize the BW manifold at an arbitrary base point (R\in\mathcal{P}_d). The linearization yields an inner product on the tangent space at (R), which in turn defines a natural distance (B_R(P,Q)=\operatorname{Tr}