Bayesian Gill-Massar Bound: An Attainable Lower Bound for Qubit Parameter Estimation

• URL: http://arxiv.org/abs/2607.07031v1
• Date: Wed, 08 Jul 2026 06:04:50 GMT
• Title: Bayesian Gill-Massar Bound: An Attainable Lower Bound for Qubit Parameter Estimation
• Abstract summary: We introduce a new lower bound, referred to as the Bayesian Gill-Massar (B-GM) bound, inspired by the logic of Gill-Massar bound in point estimation.<n>We prove that the optimal Bayesian strategy can be realized by a projection-valued measure associated with a single effective direction.<n>Our results show that the B-GM bound has a limitation in high-dimensional models with few parameters, since it can be negative.
• Score: 4.669017653337737
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study lower bounds for Bayesian quantum parameter estimation, with a particular focus on qubit models. While several lower bounds on the Bayes risk have been proposed, including the Bayesian symmetric logarithmic derivative (B-SLD) type bound and the Bayesian Nagaoka-Hayashi (B-NH) bound, there is no definite proof available to show they are attainable except for special cases. Thus, identifying attainable bounds together with their corresponding optimal measurement strategies remains a central open problem in Bayesian quantum estimation. In this work, we introduce a new Bayesian lower bound, referred to as the Bayesian Gill-Massar (B-GM) bound, inspired by the logic of Gill-Massar bound in point estimation. We derive an analytical closed-form expression of the bound and show that it is attainable for any qubit model. In particular, we prove that the optimal Bayesian strategy can be realized by a projection-valued measure associated with a single effective direction determined by the weight matrix and the B-SLD-type Fisher information matrix. We provide numerical comparisons between the B-GM, B-NH, and B-SLD-type bounds in higher-dimensional models. Our results show that the B-GM bound has a limitation in high-dimensional models with few parameters, since it can be negative.

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