Bayesian monotone metrics extend Petz metrics to prior-averaged states, yielding computable lower bounds on multiparameter Bayes risk that dominate van Trees bounds and can be optimized via a one-parameter subfamily.
Theorem 7(Quantum posterior variance).Given any operator monotone functionf, the Bayesian MSE is bounded by Quantum posterior variance VB[ˆΠ]≥M−K f B
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Bayesian Monotone Metrics for Multiparameter Quantum Estimation
Bayesian monotone metrics extend Petz metrics to prior-averaged states, yielding computable lower bounds on multiparameter Bayes risk that dominate van Trees bounds and can be optimized via a one-parameter subfamily.