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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 $\alpha$., Comment: 53 (38 + 15) pages, 3 figures. Preliminary version; comments welcome

Fidelity is arguably the most popular figure of merit in quantum sciences. However, many of its properties are still unknown. In this work, we resolve the open problem of maximizing average fidelity over arbitrary finite ensembles of quantum states and derive new upper bounds. We first construct a semidefinite program whose optimal value is the maximum average fidelity and then derive fixed-point algorithms that converge to the optimal state. The fixed-point algorithms outperform the semidefinite program in terms of numerical runtime. We also derive expressions for near-optimal states that are easier to compute and upper and lower bounds for maximum average fidelity that are exact when all the states in the ensemble commute. Finally, we discuss how our results solve some open problems in Bayesian quantum tomography., Comment: 24 + 1 pages and 5 figures. Code available at https://github.com/afhamash/optimal-average-fidelity