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We study Haar-random orthonormal bases and pretty good measurement asmeasurement choices for Bayesian state estimation. We consider differentensembles of states under an $N$-updates Bayesian algorithm given $N$Haar-random measurement bases. We obtain a bound on fidelity averaged over IIDsequences of such random measurement bases for a uniform ensemble of purestates. We also find that Clifford unitaries can only give a weak lower boundfor average fidelity in contrast to Haar random unitaries for ensembles ofmixed qubit states. For a single-shot-update, we argue using the Petz recoverycorrespondence for pretty good measurement that it can give pretty goodBayesian mean estimates.

Haar-random and pretty good measurements for Bayesian state estimation