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 (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.

Global Quantum Thermometry

“…In contrast, a local analysis can lead to a biased temperature estimate even for large μ. These results show that a paradigm shift toward Bayesian techniques may allow a more robust and significantly enhanced optimization of thermometric protocols, especially in cases where the data are limited [23,27,28].…”

Quantum Fisher information maximization in an unbalanced interferometer

“…Appendix D: The Fisher matrix coefficient F sd Employing relation (C1), after some straightforward algebra we are led to the result from equation (20). Please note that we used the covariance (21) and not the symmetrized covariance (18) because N commutes with both Ĵy and Ĵz . By direct calculation one finds…”

“…Before moving on, one must mention other noteworthy approaches for optimal parameter estimation, including the Bayesian method [9,21] or boson-sampling inspired strategies [22].…”

Quantum Scale Estimation

“…Despite the generality of Eq. ( 1), it is often useful to summarise the key features of the posterior density via a point estimator θ(x) with uncertainty [56,73,75,76]…”

“…[33], but, notably, it also provides a new rule to calculate the associated probability-operator measurement (POM) 2 that is optimal for any amount of prior knowledge and a given quantum state. Built on Bayesian principles 3 [24,59], this framework is universally valid, with no restrictions on numbers of resources or initial information [46,73].…”