Optimizing quantum-enhanced Bayesian multiparameter estimation in noisy apparata

Abstract

Achieving quantum-enhanced performances when measuring unknown quantities requires developing suitable methodologies for practical scenarios, that include noise and the availability of a limited amount of resources. Here, we report on the optimization of quantum-enhanced Bayesian multiparameter estimation in a scenario where a subset of the parameters describes unavoidable noise processes in an experimental photonic sensor. We explore how the optimization of the estimation changes depending on which parameters are either of interest or are treated as nuisance ones. Our results show that optimizing the multiparameter approach in noisy apparata represents a significant tool to fully exploit the potential of practical sensors operating beyond the standard quantum limit for broad resources range.

Benchmarking Bayesian quantum estimation

“…In this case, the mean is greater than the median, and the proportionality factor among them can be estimated numerically computing the ratio of the median of a generic squared Gaussian distribution centered in zero and its mean. This factor, in the single-parameter case, results equal to k ≃ 0.4549 [25] and it must be taken into consideration when comparing the performances in terms of medians with the ultimate precision bound. Moreover, we underline that when comparing the overall estimation performances to the precision bound, after having considered either the mean or the median over the different repetitions of the estimate (that we indicate respectively with M[•] and M[•]), it is necessary to consider the average performances among all the different parameter values.…”

“…As previously discussed, our approach to computing the necessary integrals for Bayesian estimation involved discretizing the parameter space into n possible values covering the entire periodicity interval. This discretization was conducted in accordance with SMC, also known as the particle filtering method [44], used in recent multiparameter estimation experiments [24][25][26]. The objective was to reconstruct the posterior probability distribution of the parameter and then use it to estimate its value.…”

“…The realm of possible adaptive techniques is varied. Among the others, it includes machine learning-based algorithms [27,52], variational techniques [53,54], Bayesian updates based on minimizing the variance of the posterior [24][25][26], particle swarm optimization [19], genetic algorithms [23], or differential evolution [20,55]. The choice of one of these efficient adaptive protocols, involving a feedback loop, becomes crucial to achieve the ultimate sensitivity in multiparameter scenarios.…”

“…This optimization process can be done through different online and offline [19][20][21][22][23] protocols. One of the most implemented ones is developed within the Bayesian inference framework, and is based on updating the knowledge of the parameter value depending on the registered measurement outcome and on the setting of some control parameter [24][25][26][27] valid for both the qubit framework [28] and the continuous-variable one [29]. The advantage becomes particularly significant in the multiparameter framework where the interest relies on the simultaneous estimation of several parameters [30][31][32][33].…”