Saturating the Bayesian Nagaoka-Hayashi bound within numerical precision for the depolarization SU(2) rotation channel

• URL: http://arxiv.org/abs/2607.15398v1
• Date: Thu, 16 Jul 2026 18:56:31 GMT
• Title: Saturating the Bayesian Nagaoka-Hayashi bound within numerical precision for the depolarization SU(2) rotation channel
• Abstract summary: We study the single-shot estimation of a qubit rotation, all three parameters at once, under depolarizing noise and a uniform prior.<n> Rotational covariance reduces the joint optimization of probe, measurement, and estimator, and the NH bound itself, to small semidefinite programs.<n>At every number of uses and noise strength the optimized strategy reaches the NH bound within the numerical precision of the calculation.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Bayesian Nagaoka--Hayashi (NH) bound is a semidefinite lower bound on the Bayes risk of multiparameter estimation, tighter than the Bayesian symmertic logarithmic derivative (SLD) Cramér--Rao bounds, and whether it can be attained is an open problem. We study the single-shot estimation of a qubit rotation, all three parameters at once, under depolarizing noise and a uniform prior, with $k$ parallel uses of the channel. Rotational covariance reduces the joint optimization of probe, measurement, and estimator, and the NH bound itself, to small semidefinite programs, solvable through $k=4$. At every number of uses and noise strength the optimized strategy reaches the NH bound within the numerical precision of the calculation, numerical evidence that the bound is attained for this channel family, while the SLD bound lies strictly below and is never attained. As the noise grows the optimal probe collapses to the tensored Bell state.

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