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

• URL: http://arxiv.org/abs/2410.04937v1
• Date: Mon, 7 Oct 2024 11:28:26 GMT
• Title: Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance
• Authors: A. Afham, Chris Ferrie,
• Abstract summary: generalizes standard quantum fidelities such as Uhlamnn-, Holevo-, and Matsumoto fidelity. We provide a Block-matrix characterization and prove an Uhlmann-like theorem.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a family of fidelities based on the Riemannian geometry of the Bures-Wasserstein manifold we call the generalized fidelity. We show that this family of fidelities generalizes standard quantum fidelities such as Uhlamnn-, 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 from the linearization of 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 quantum Renyi divergences, generalizing Petz-, Sandwich-, Reverse sandwich-, and Geometric Renyi divergences of order $\alpha$.

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