Bayesian and frequentist estimators for the transition frequency of a driven two-level quantum system

TL;DR

This paper compares Bayesian and frequentist methods for estimating the transition frequency of a driven two-level quantum system, highlighting the advantages of Bayesian estimators in providing unambiguous results.

Contribution

It introduces a cost function to optimize Fisher information and analyzes the effectiveness of Bayesian versus frequentist estimators in quantum parameter estimation.

Findings

Bayesian estimators yield unambiguous estimates.

Frequentist estimators can be ambiguous or undefined due to nonlinearity.

The proposed cost function improves classical Fisher information optimization.

Abstract

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 a posteriori 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.