Fig. 6: Experimental results.
From: Experimental adaptive Bayesian estimation of multiple phases with limited data
Simultaneous estimations of Nph = 15 different pairs of phases using Bayesian adaptive protocol. Quadratic loss is averaged for each phase over \({N}_{\exp }=100\) independent runs. a Comparison between overall quadratic loss \(L({\boldsymbol{\phi} },\hat{{\boldsymbol{\phi} }})\) (red dots) and \({\rm{Tr}}[{({{\mathcal{F}}}_{\exp })}^{-1}]/N\) (red solid line). The performances are in agreement with the numerical simulations. Red shaded regions represents the one-standard deviation interval where \(L({\boldsymbol{\phi} },\hat{{\boldsymbol{\phi} }})\) can be found. b Analysis of diagonal elements of CRB by comparing quadratic loss relative to the single phase of the estimated pair \(L({\phi }_{i},\hat{{\phi }_{i}})\) (with i = 1, 2) (blue triangles) and \({\rm{Tr}}[{({{\mathcal{F}}}_{\exp })}^{-1}]/(2N)\) (blue solid line). The algorithm shows symmetric optimal performances for estimation of both parameters, by using the same amount of resources. This feature is highlighted by the inset panel, where the ratio \(\Delta L({\boldsymbol{\phi} },\hat{{\boldsymbol{\phi} }})\) between the difference of the two estimations and the bound value is reported. c, Analysis of phase correlations by comparing off-diagonal terms of \({{\mathcal{F}}}_{\exp }^{-1}\) (see Supplementary Note 4) and \({[{({{\mathcal{F}}}_{\exp })}^{-1}]}_{12}/N\) (green solid line). d Estimation of convergence time (τN) to CRB. The value can be estimated by fitting the distance between the averaged \(L({\boldsymbol{\phi} },\hat{{\boldsymbol{\phi} }})\) and CRB, after N > 2. The adopted fit function is \(a+b\exp (-N/{\tau }_{N})\), with \(a,b,{\tau }_{N}\in {\mathbb{R}}\) the fitting parameters, leading to τN = 5.6. The choice of this function is performed to provide a reasonable estimation of τN, as the number of probes necessary to approach the CRB.
