Influence of parameter prior information on effect of colored noise in Bayesian frequency estimation

Parameter estimation, which undertakes one of the vital missions in quantum metrology, has attracted a lot of attention in recent years. A large number of investigations on the frequency estimation have been carried out. Most of them are based on Cramér-Rao bound estimation approach in which almost perfect knowledge of the parameter to be estimated is given. In reality, however, one has inadequate prior knowledge about the parameter to be estimated. Then the Bayesian estimation approach in which we can perform the estimation even if we only have partial prior information about the parameter would be an ideal choice. Prior information about the parameter can play a significant role in Bayesian statistical inference. So it is interesting to know how the prior knowledge affects the estimation accuracy in the estimation process. In the solid-state realization of probe system, material-specific fluctuations typically lead to the major contribution to the intrinsic noise. Then it is interesting to study the effects of colored noise on the quantum parameter estimation. In this work, we study the inhibitory effects of prior probability distribution of the parameter to be estimated on the effects of colored noise under the framework of Bayesian parameter estimation theory. In particular, we estimate the intensity of a magnetic field by adopting a spin-1/2 system which is influenced by the colored noise with 1/f^α spectrum. To evaluate the accuracy of estimation, we obtain the Bayes cost analytically which can be applied to the noisy channels. We mainly focus on the inhibitory effect of prior probability distribution of measured parameter on the non-Gaussianity of noise. We find that for the case of broad prior frequency distribution, the influence of non- Gaussianity on the estimation is very weak. While for the case of narrow prior frequency distribution, the influence of non-Gaussianity on the estimation is strong. That means that in the Bayesian approach, when we have enough prior information about the frequency, the non-Gaussianity can conduce to the improvement of the accuracy of the estimation of the frequency. When we lose the prior information, we also lose the improvement of the accuracy from the non-Gaussianity. The uncertainty of the prior information tends to eliminate the effects of the non-Gaussianity of the noise.