We benchmark the relative ordering of three important lower bounds on the Bayesian mean-square error: Bayesian quantum Cramer-Rao bound (BQCRB), non-Hermitian quantum Weiss-Weinstein bound (QWWB), and quantum Ziv-Zakai bound (QZZB) for noisy frequency estimation with a spin-1/2 under pure dephasing. We find that both the noise characteristics and the prior distribution of the unknown frequency have strong influence on the ordering of these bounds: narrow prior distribution and/or strong noise favor the ordering (i): $\text{BQCRB}>\text{QWWB}>\text{QZZB}$, otherwise the ordering (ii): $\text{QWWB}>\text{QZZB}>\text{BQCRB}$ is favored. Interestingly, QWWB is worse than BQCRB when the noise is strong, in contrast to the noiseless case where QWWB is never worse than BQCRB. Another interesting finding is that increasing the non-Markovianity of the noise usually makes ordering (i) more favorable. This work underscores the critical role of noise characteristics and prior information in the selection of these Bayesian bounds for quantum metrology in noisy environments.

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• Quantum metrology
• Quantum parameter estimation