Chun Kit Dennis Law | Searcharxiv

arXiv2026

Bayesian approach to simultaneous quantum multiparameter estimation with finite data

Multiparameter estimation remains a fundamental challenge in quantum estimation theory because the optimal measurements associated with different parameters are generally incompatible. In this work, we develop a Bayesian framework for the simultaneous estimation of multiple parameters. Our approach builds on Personick's method, in which the estimation problem is reduced to a set of Lyapunov equations whose solutions define optimal observables for the individual parameters. Since these observables generally do not commute, we construct a convex combination of the solutions, parameterized by a set of variational parameters. The spectral decomposition of the resulting operator defines a parametrized projection-valued measure. The resulting projective measurement determines the likelihood function, from which the minimum mean-square error estimators and the corresponding Bayesian mean-square errors are obtained, following standard Bayesian procedures, as functions of the variational parameters. To determine their optimal values, we formulate a minimax optimization problem based on the normalized Bayesian mean-square errors, introducing a min-max normalization procedure that enables a meaningful comparison of estimation errors associated with different parameters. This optimization yields a variationally optimized projective measurement for simultaneous Bayesian estimation. Two qubit examples, involving phase estimation and the estimation of parameters defining convex combinations of unitary operations, demonstrate the construction of optimized projective measurements and the corresponding minimum mean-square error estimators within the proposed framework for finite data sets.

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arXiv2025

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

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.

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