Bayesian quantum estimation provides a robust formalism for quantum technologies, particularly in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable systems has remained limited and largely numerical due to the difficulty of the underlying parameter integrals. Here, we introduce a variational framework that restricts the optimisation over all measurements and estimators to a finite-dimensional subspace, reducing the problem to a linear system of equations with closed-form solutions. These solutions have a geometric interpretation as orthogonal projections of the global optimum onto the chosen subspace. For Gaussian states, a natural subspace is spanned by operators polynomial in the canonical quadratures, whose moments are computable from the first and second moments of the state. We further derive a necessary and sufficient condition for global optimality, and show that post-processing with the posterior mean, rather than using the estimator obtained from the subspace, improves performance towards the global optimum. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal.
Bayesian quantum estimation provides a robust formalism for quantum technologies, particularly in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable systems has remained limited and largely numerical due to the difficulty of the underlying parameter integrals. Here, we introduce a variational framework that restricts the optimisation over all measurements and estimators to a finite-dimensional subspace, reducing the problem to a linear system of equations with closed-form solutions. These solutions have a geometric interpretation as orthogonal projections of the global optimum onto the chosen subspace. For Gaussian states, a natural subspace is spanned by operators polynomial in the canonical quadratures, whose moments are computable from the first and second moments of the state. We further derive a necessary and sufficient condition for global optimality, and show that post-processing with the posterior mean, rather than using the estimator obtained from the subspace, improves performance towards the global optimum. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal.
Bayesian analysis is a framework for parameter estimation that applies even in uncertainty regimes where the commonly used local (frequentist) analysis based on the Cramer-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.
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.
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.
We propose a machine learning framework for parameter estimation of single mode Gaussian quantum states. Under a Bayesian framework, our approach estimates parameters of suitable prior distributions from measured data. For phase-space displacement and squeezing parameter estimation, this is achieved by introducing Expectation-Maximization (EM) based algorithms, while for phase parameter estimation an empirical Bayes method is applied. The estimated prior distribution parameters along with the observed data are used for finding the optimal Bayesian estimate of the unknown displacement, squeezing, and phase parameters. Our simulation results show that the proposed algorithms have estimation performance that is very close to that of Genie Aided Bayesian estimators, that assume perfect knowledge of the prior parameters. In practical scenarios, when numerical values of the prior distribution parameters are not known beforehand, our proposed methods can be used to find optimal Bayesian estimates from the observed measurement data.
Abstract The quest for precision in parameter estimation is a fundamental task in different scientific areas. The relevance of this problem thus provided the motivation to develop methods for the application of quantum resources to estimation protocols. Within this context, Bayesian estimation offers a complete framework for optimal quantum metrology techniques, such as adaptive protocols. However, the use of the Bayesian approach requires extensive computational resources, especially in the multiparameter estimations that represent the typical operational scenario for quantum sensors. Hence, the requirement to characterize protocols implementing Bayesian estimations can become a significant challenge. This work focuses on the crucial task of robustly benchmarking the performances of these protocols in both single and multiple-parameter scenarios. By comparing different figures of merits, evidence is provided in favor of using the median of the quadratic error in the estimations in order to mitigate spurious effects due to the numerical discretization of the parameter space, the presence of limited data, and numerical instabilities. These results, providing a robust and reliable characterization of Bayesian protocols, find natural applications to practical problems within the quantum estimation framework.
Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances of the problem, but in general even numerical methods are not known. Here, we focus on the single-shot Bayesian setting, where the goal is to find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024