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Physical Review A

Effect of prior information on noisy Bayesian frequency estimation

Jungang Li, Yuanmei Wang, Di Yang, Jian Zou

Published Nov 27, 2017• 4 citations
Publisher

Abstract

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 Cram\'er-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.

DOI
10.1103/physreva.96.052130

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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 Cram'er-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.
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/fα 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.
We advocate a Bayesian approach to optimal quantum frequency estimation—an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise, the average Bayesian cost can be expressed using the quantum Fisher information. Thus we establish a direct link between the two, often competing, approaches to quantum estimation theory.
We advocate a Bayesian approach to optimal quantum frequency estimation—an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise, the average Bayesian cost can be expressed using the quantum Fisher information. Thus we establish a direct link between the two, often competing, approaches to quantum estimation theory.
Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér-Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér-Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.
We advocate a Bayesian approach to optimal quantum frequency estimation - an important problem for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of the prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing the local oscillator noise as well as additional atomic dephasing effects. We also demonstrate an important fact: the average Bayesian cost can be expressed explicitly using the quantum Fisher information and thus establish a direct link between the two, often competing, approaches to quantum estimation theory.
Abstract Bayesian analysis is a framework for parameter estimation that applies even in uncertainty regimes where the commonly used local (frequentist) analysis based on the Cramér–Rao bound (CRB) is not well defined. In particular, it applies when no initial information about the parameter value is available, e.g., when few measurements are performed. Here, we consider three paradigmatic estimation schemes in continuous-variable (CV) quantum metrology (estimation of displacements, phases, and squeezing strengths) and analyse them from the Bayesian perspective. For each of these scenarios, we investigate the precision achievable with single-mode Gaussian states under homodyne and heterodyne detection. This allows us to identify Bayesian estimation strategies that combine good performance with the potential for straightforward experimental realization in terms of Gaussian states and measurements. Our results provide practical solutions for reaching uncertainties where local estimation techniques apply, thus bridging the gap to regimes where asymptotically optimal strategies can be employed.