On the Parameter Estimation Problem for Quantum Statistical Models

Hiroshi Nagaoka

Sholeh Razavian, Matteo G. A. Paris, Marco G. Genoni

The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumnessR, a measure that has been recently introduced in order to quantify how incompatible the parameters to be estimated are. In particular, R is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cramér-Rao bound (i.e., the matrix generalization of the single-parameter quantum Cramér-Rao bound). For all the estimation problems considered, we evaluate the quantumness R and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that R is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model, it is equal to the difference between the bounds and, in general, their behavior qualitatively coincide. On the other hand, we also find evidence that, for certain quantum statistical models, the bound is not in tight, and thus R may overestimate the degree of quantum incompatibility between parameters.

Jun Suzuki

Abstract In this paper, we analyze quantum-state estimation problems when some of the parameters are of no interest to be estimated. In classical statistics, these irrelevant parameters are called nuisance parameters, and this problem is of great importance in many practical applications of statistics. However, little is known regarding the effects of nuisance parameters in quantum-state estimation problems. The main contribution of this paper is first to formulate the nuisance parameter problem in the quantum-state estimation theory for the finite sample size case. We then derive the precision limit for the parameters of interest based on the concept of locally unbiasedness for the parameters of interest. We also propose a method of how to eliminate the nuisance parameters to obtain an upper bound for estimation error of the parameters of interest. Based on the proposed method, we clarify an intrinsic tradeoff relation to estimate the nuisance parameters and parameters of interest. The general qubit models are examined in detail to emphasize that we cannot ignore the effects of the nuisance parameters in general. Several examples are worked out to illustrate our findings.

浩一 山形

Alexander Semenovich Holevo

Tomoki Tanaka, Shumpei Uno, Tamiya Onodera +2 more

Many quantum algorithms contain an important subroutine---the quantum amplitude estimation. As the name implies, this is essentially the parameter estimation problem and thus can be handled via the established statistical estimation theory. However, this problem has an intrinsic difficulty that the system, i.e., the real quantum computing device, inevitably introduces unknown noise; the probability distribution model then has to incorporate many nuisance noise parameters, resulting that the construction of an optimal estimator becomes inefficient and difficult. For this problem we apply the theory of nuisance parameters (more specifically, the parameter orthogonalization method) to precisely compute the maximum likelihood estimator for only the target amplitude parameter by removing the other nuisance noise parameters. That is, we can estimate the amplitude parameter without estimating the noise parameters. We validate the parameter orthogonalization method in a numerical simulation and study the performance of the estimator in the experiment using a real superconducting quantum device.