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.
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.
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.
The frequentist (local) quantum parameter estimation based on the Cramer-Rao bound is an asymptotic theory and therefore requires an infinite number of samples for the measurements. On the other hand, by introducing the prior probability distribution, Bayesian parameter estimation approach relaxes this requirement and can be successfully applied under realistic assumption of possessing a limited prior information about the parameter value under investigation. Meanwhile, recent advances in PT-symmetric quantum mechanics, both theoretically and experimentally, provide an opportunity to significantly enhance the parameter estimation precision by manipulating the Hilbert space of the system. Taking advantage of these, we develop the Bayesian parameter estimation approach for the two-level system possessing PT symmetry. We consider both options available in PT-symmetric quantum mechanics, namely the CPT and Hermitian measurements, for encoding the probe state and manipulation of the quantum Fisher information of the system and take into account possible decoherence effects inherent for the practical implementations. Our approach should be particularly relevant for applications where the accuracy of quantum parameter estimation is crucial, such as quantum randomness sources and the quantum key distribution. Finally, to bridge the gap with the practical implementations, we consider the relation of our scheme with the current implementation of PT symmetry on the IBM quantum processor and formulate our approach as Bayesian parameter estimation employing an ancilla qubit.
The formalism of quantum estimation theory with a specific focus on classical data postprocessing is applied to a two-level system driven by an external gyrating magnetic field. We employed both Bayesian and frequentist approaches to estimate the unknown transition frequency. In the frequentist approach, we have shown that only reducing the distance between the classical and the quantum Fisher information does not necessarily mean that the estimators as functions of the data deliver an estimate with desirable accuracy, as the classical Fisher information takes small values. We have proposed and investigated a cost function to account for the maximization of the classical Fisher information and the minimization of the aforementioned distance. Due to the nonlinearity of the probability mass function of the data on the transition frequency, the minimum variance unbiased estimator may not exist. The maximum likelihood and the maximum estimators often result in ambiguous estimates, which in certain cases can be made unambiguous upon changing the parameters of the external field. It is demonstrated that the minimum mean-square error estimator of the Bayesian statistics provides unambiguous estimates. In the Bayesian approach, we have also investigated the effects of noninformative and informative priors on the Bayesian estimates, including a uniform prior, Jeffrey's prior, and a Gaussian prior.
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.
Parameter estimation, especially frequency estimation, from noisy observations of interference is essential in optical interferometric sensing and metrology. The Cramer-Rao bound (CRB) of such estimation determines measurement sensitivity limit. Unlike the well-studied complex sinusoids in communication theory, an optical interference signal is distinctly different in its model parameters and noise statistics. The connection between these parameters and their estimation bounds has not been well understood. Here we propose a complete, realistic multiparameter interference model corrupted by a combination of shot noise, dark noise, and readout noise. We derive the Fisher information matrix and the CRBs for all model parameters, including intensity, visibility, optical path length (frequency), and initial phase. We show that the CRBs of frequency and phase are coupled but not affected by the knowledge of intensity and visibility. Knowing the initial phase offers significant sensitivity advantage, which is verified by both theoretical derivations and numerical simulations. In addition to the complete model, a shot noise-limited case is studied, permitting the calculation of the CRBs directly from measured data.
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.