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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 theopen problem of maximizing average fidelity over arbitrary finite ensembles ofquantum states and derive new upper bounds. We first construct a semidefiniteprogram whose optimal value is the maximum average fidelity and then derivefixed-point algorithms that converge to the optimal state. The fixed-pointalgorithms outperform the semidefinite program in terms of numerical runtime.We also derive expressions for near-optimal states that are easier to computeand upper and lower bounds for maximum average fidelity that are exact when allthe states in the ensemble commute. Finally, we discuss how our results solvesome open problems in Bayesian quantum tomography.

Quantum mean states are nicer than you think: fast algorithms to compute states maximizing average fidelity