Bayesian Monotone Metrics for Multiparameter Quantum Estimation

• URL: http://arxiv.org/abs/2607.01685v1
• Date: Thu, 02 Jul 2026 04:27:28 GMT
• Title: Bayesian Monotone Metrics for Multiparameter Quantum Estimation
• Abstract summary: We introduce Bayesian monotone metrics by evaluating Petz monotone metrics on the prior-averaged state.<n>For every monotone metric in the family, we prove a universal dominance over the corresponding quantum van Trees.<n>Our results establish Bayesian monotone metrics as a unifying information-geometric perspective on Bayesian quantum estimation.
• Score: 4.669017653337737
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Bayesian quantum estimation offers a finite-data framework for quantum sensing and metrology, yet a unified geometric formulation for multiparameter Bayes risk has been lacking. We introduce Bayesian monotone metrics by evaluating Petz monotone metrics on the prior-averaged state, providing a Bayesian extension of the full class of statistically meaningful (CPTP) quantum metrics. This framework yields Bayesian quantities, including quantum posterior-mean operators and a quantum Bayesian dual Fisher-information matrix, and it leads to a systematic family of computable lower bounds on the Bayes risk. The resulting bounds naturally incorporate multiparameter measurement incompatibility and, for every monotone metric in the family, we prove a universal dominance over the corresponding quantum van Trees (Bayesian Cramér--Rao) bound. Moreover, we show that optimizing over all operator monotone functions collapses to a one-parameter subfamily, turning the tightest bound into a tractable optimization with a clear geometric interpretation. In representative examples, the optimized bounds are strictly tighter than the Bayesian SLD and RLD bounds. Our results establish Bayesian monotone metrics as a unifying information-geometric perspective on Bayesian quantum estimation, enabling systematic and computable performance limits in multiparameter settings.

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