Effect of prior information on noisy Bayesian frequency estimation

Quantum frequency estimation under general noise is investigated in the Bayesian parameter estimation approach. To evaluate the accuracy of estimation, the Bayes cost is obtained analytically which can be applied to the common noisy channels, such as the phase-damping channel, the amplitude-damping channel, and the depolarizing channel. The Bayes cost formula clearly shows that the prior information imposes a restriction upon the effect of noise in the estimation process. Three examples of frequency estimations are provided to illustrate the roles of prior probability in the estimation process. It is found that, due to the restriction of prior information on the noise, the estimation accuracy is less sensitive to the noise in the Bayesian approach than that in the Cramér-Rao bound approach. More prior information can help us to use the non-Gaussianity of the noise channel to improve the estimation accuracy in the phase-damping channel.