Bayesian error regions in quantum estimation I: analytical reasonings

Abstract

Results concerning the construction of quantum Bayesian error regions as a means to certify the quality of parameter point estimators have been reported in recent years. This task remains numerically formidable in practice for large dimensions and so far, no analytical expressions of the region size and credibility (probability of any given true parameter residing in the region) are known, which form the two principal region properties to be reported alongside a point estimator obtained from collected data. We first establish analytical formulas for the size and credibility that are valid for a uniform prior distribution over parameters, sufficiently large data samples and general constrained convex parameter estimation settings. These formulas provide a means to an efficient asymptotic error certification for parameters of arbitrary dimensions. Next, we demonstrate the accuracies of these analytical formulas as compared to numerically computed region quantities with simulated examples in qubit and qutrit quantum-state tomography where computations of the latter are feasible.

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

“…For the purpose of laying out the foundations for subsequent discussion on region accuracy and adaptive quantum estimation, we state the key properties of a Bayesian credible-region   = l that is characterized by 0λ1 with an isolikelihood boundary. The size and credibility of  l are defined in (1) of [2]. From hereon, we shall focus(see later section 3.5) on the situation where the true parameter r  Ï ¶ , so that for a sufficiently large data sample size N, the error region  for all interesting values of λ has boundary…”

“…that measures the region accuracy relative to r, or the average accuracy of all the points in , where r d ¢ ( )is the normalized integral measure as defined in [2]. It is easy to see that when r…”

“…The preceding companion article [2] discussed asymptotic techniques for constructing Bayesian regions for the maximum-likelihood (ML) estimator. Such a Bayesian region annotates the estimator with credibility that it lies in this region of a given size.…”

“…In the limit of zero region size, the region accuracy becomes the usual point-estimator accuracy. After a review in section 2 on Bayesian regions, we shall see in section 3 that this notion of region accuracy is intimately connected to the dual operations [1,2] of fixing the region size while increasing credibility, or fixing the credibility-both actions tend to increase the region accuracy, and this tendency becomes exact in single-parameter quantum estimation.…”

“…These schemes will be applied to three examples in quantum estimation that can be categorized under quantum metrology and Gaussian state characterization. All symbols and notations from [2] are carried over to this article. The prior distribution for the true parameter shall again be taken to be the uniform primitive distribution in the parameter space.…”