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.
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.
Abstract In parameter estimation, nuisance parameters refer to parameters that are not of interest but nevertheless affect the precision of estimating other parameters of interest. For instance, the strength of noises in a probe can be regarded as a nuisance parameter. Despite its long history in classical statistics, the nuisance parameter problem in quantum estimation remains largely unexplored. The goal of this article is to provide a systematic review of quantum estimation in the presence of nuisance parameters, and to supply those who work in quantum tomography and quantum metrology with tools to tackle relevant problems. After an introduction to the nuisance parameter and quantum estimation theory, we explicitly formulate the problem of quantum state estimation with nuisance parameters. We extend quantum Cramér–Rao bounds to the nuisance parameter case and provide a parameter orthogonalization tool to separate the nuisance parameters from the parameters of interest. In particular, we put more focus on the case of one-parameter estimation in the presence of nuisance parameters, as it is most frequently encountered in practice.
In many sensing applications, the quantity of interest is not the only unknown, there are also additional unknown parameters, known as nuisance parameters, that affect the precision of estimation. While the ultimate local precision limit for a target parameter is well understood in the absence of nuisance parameters, the problem becomes significantly more challenging when they are present. In this work, we develop a framework for optimal Hamiltonian parameter estimation in the presence of nuisance parameters. We introduce an effective generator that captures the influence of nuisance parameters on the target precision, providing an explicit characterization of the ultimate precision limit for estimating the target parameter. Finally, we provide explicit optimal protocols, including probe state, control, and measurement that saturate this fundamental limit.
We derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non-degenerate. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator’s deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a useful asymptotic characterization of identically prepared states in terms of Gaussian states. We first prove our results for the mean square error of a special class of models, called D -invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, i.e. parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.
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.
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.
The quantum Cramer-Rao bound places a limit on the mean square error of a parameter estimation procedure, and its numerical value is determined by the quantum Fisher information. For single parameters, this leads to the well- known Heisenberg limit that surpasses the classical shot-noise limit. When estimating multiple parameters, the situation is more complicated and the quantum Cramer-Rao bound is generally not attainable. In such cases, the use of entanglement typically still offers an enhancement in precision. Here, we demonstrate that entanglement is detrimental when estimating some nuisance parameters. In general, we find that the estimation of coupled parameters does not benefit from either classical or quantum correlations. We illustrate this effect in a practical application for optical gyroscopes.