Haar-random and pretty good measurements for Bayesian state estimation

Maria Quadeer

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arXiv · 2310.20565

Haar-random and pretty good measurements for Bayesian state estimation

Maria Quadeer

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Abstract

We study Haar-random bases and pretty good measurement for Bayesian state estimation. Given $N$ Haar-random bases we derive a bound on fidelity averaged over IID sequences of such random measurements for a uniform ensemble of pure states. For ensembles of mixed qubit states, we find that measurements defined through unitary 2-designs closely approximate those defined via Haar random unitaries while the Pauli group only gives a weak lower bound. For a single-shot-update, we show using the Petz recovery map for pretty good measurement that it can give pretty good Bayesian mean estimates.

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Maria Quadeer. 2023-10-31. Haar-random and pretty good measurements for Bayesian state estimation. https://arxiv.org/abs/2310.20565

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