Bayesian error regions in quantum estimation II: region accuracy and adaptive methods

Abstract

Bayesian error analysis paves the way to the construction of credible and plausible error regions for a point estimator obtained from a given dataset. We introduce the concept of region accuracy for error regions (a generalization of the point-estimator mean squared error) to quantify the average statistical accuracy of all region points with respect to the unknown true parameter. We show that the increase in region accuracy is closely related to the Bayesian region dual operations in Shang et al (2013 New J. Phys. 15 123026). Next with only the given dataset as viable evidence, we establish various adaptive methods to maximize the region accuracy relative to the true parameter subject to the type of reported Bayesian region for a given point estimator. We highlight the performance of these adaptive methods by comparing them with non-adaptive procedures in three quantum-parameter estimation examples. The results of and mechanisms behind the adaptive schemes can be understood as the region analog of adaptive approaches to achieving the quantum Cramér-Rao bound for point estimators.

Bayesian error regions in quantum estimation I: analytical reasonings

“…For this prior, the volume for the (d=D 2 −1)-dimensional state parameter r has the closed form [22] in terms of the (d=3)-dimensional state parameter r. The qubit space also has the nice property that the boundary  ¶ 2 is smooth-it is the surface of a 2-sphere. This implies that  ¶ 2 is smooth and can eventually be described by a hyperplane for sufficiently large N. We shall see that the expressions in (11), (12), (14) and (15) indeed exactly describe the actual size and credibility in this limit.…”

“…This allows any observer to conduct asymptotic error certification for uniform priors that avoids NPhard Monte Carlo computations. The theory supplies analytical formulas for the region size and credibility in cases where the true parameter is an interior point (equation (7), (10), (11) and (12)), as well as the case where the true parameter is on the boundary of the parameter space (equation (14) and (15)). These expressions approach the exact answers whenever the joint boundary of both the region and full parameter space is smooth.…”

“…In all the Bayesian region property formulas developed ((7), (10), (11), (12), (14), (15)) as a means to provide an asymptotic size and credibility certification for the ML estimator  r ML , the size formulas exhibit logarithmic divergences-l l ( ) s log d 2 . This feature stems from the Gaussian approximations in (5) and (13) that pays no attention to the parameter space boundary    ¶ ¶ Ç ¶ ⧹( ) 0 0 that falls on 'the other side' of the joint one (if there is any).…”

“…Figure 9 studies the behaviors of theoretical results for these two cases. 7 In [14], where we study single-phase estimation with Bayesian regions for a different purpose, Case2 shall apply to an enforced uniform prior that covers a subset of the phase interval since the likelihood at the boundary points can be nonzero in this case. Figure 10 illustrates the validity of our theory.…”

“…We show, with examples of quantum-state tomography, that the expressions work well even for moderately large sample size. In the companion article [14] we shall discuss various adaptive methods that optimize tomographic accuracy in the context of these Bayesian regions.…”