Haar-random and pretty good measurements for Bayesian state estimation

2023/10/31 by Quadeer, Maria · 1 citation
Computer Science · Mathematics · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2310.20565

openalex publication_date 2023/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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