Bayesian parameter estimation using Gaussian states and measurements

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 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 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.

Measurement-Device-Independent Entanglement Detection for Continuous-Variable Systems

“…A lower bound, in turn, can be obtained using a multiparameter Bayesian version of the quantum Cramér-Rao bound [25]. The simultaneous estimation of the position and momentum quadratures has been studied thoroughly and is optimized for coherent states by measuring x and p on two different modes after a 50:50 beamsplitter [26,27]. In particular, assuming a Gaussian prior distribution, like in our case, the minimal sum of variances is equal to σ 2 /(1+σ 2 ) [26], which proves the bound for the entanglement witness.…”