Quantum parameter estimation holds the promise of quantum technologies, in which physical parameters can be measured with much greater precision than what is achieved with classical technologies. However, how to obtain a best precision when the optimal measurement is not accessible is still an open problem. In this work, we present a theoretical framework to explore the parameter estimation with limited access of measurements by analyzing the effect of non-optimal measurement on the estimation precision. We define a quantity to characterize the effect and illustrate how to optimize observables to attain a bound with limited accessibility of observables. On the other side, we introduce the minimum Euclidean distance to quantify the difference between an observable and the optimal ones in terms of Frobenius norm and find that the measurement with a shorter distance to the optimal ones benefits the estimation. Two examples are presented to show our theory. In the first, we analyze the effect of non-optimal measurement on the estimation precision of the transition frequency for a driven qubit. While in the second example, we consider a bipartite system, in which one of them is measurement inaccessible. To be specific, we take a toy model, the NV-center in diamond as the bipartite system, where the NV-center electronic spin interacts with a single nucleus via the dipole-dipole interaction. We achieve a precise estimation for the nuclear Larmor frequency by optimizing only the observables of the electronic spin. In these two examples, the minimum Euclidean distance between an observable and the optimal ones is analyzed and the results show that the observable closed to the optimal ones better the estimation precision.
Quantum parameter estimation holds the promise of quantum technologies, in which physical parameters can be measured with much greater precision than what is achieved with classical technologies. However, how to obtain a best precision when the optimal measurement is not accessible is still an open problem. In this work, we present a theoretical framework to explore the parameter estimation with limited access of measurements by analyzing the effect of non-optimal measurement on the estimation precision. We define a quantity to characterize the effect and illustrate how to optimize observables to attain a bound with limited accessibility of observables. On the other side, we introduce the minimum Euclidean distance to quantify the difference between an observable and the optimal ones in terms of Frobenius norm and find that the measurement with a shorter distance to the optimal ones benefits the estimation. Two examples are presented to show our theory. In the first, we analyze the effect of non-optimal measurement on the estimation precision of the transition frequency for a driven qubit. While in the second example, we consider a bipartite system, in which one of them is measurement inaccessible. To be specific, we take a toy model, the NV-center in diamond as the bipartite system, where the NV-center electronic spin interacts with a single nucleus via the dipole-dipole interaction. We achieve a precise estimation for the nuclear Larmor frequency by optimizing only the observables of the electronic spin. In these two examples, the minimum Euclidean distance between an observable and the optimal ones is analyzed and the results show that the observable closed to the optimal ones better the estimation precision.
Quantum parameter estimation holds the promise of quantum technologies, in which physical parameters can be measured with much greater precision than what is achieved with classical technologies. However, how to obtain the best precision when the optimal measurement is not accessible is still an open problem. In this paper, we present a theoretical framework to explore the parameter estimation with limited access of measurements by analyzing the effect of nonoptimal measurement on the estimation precision. We define a quantity $\mathrm{\ensuremath{\Lambda}}$ to characterize the effect and illustrate how to optimize observables to attain a bound with limited accessibility of observables. In addition, we introduce the minimum Euclidean distance to quantify the difference between an observable and the optimal ones in terms of the Frobenius norm and find that the measurement with a shorter distance to the optimal ones benefits the estimation. Two examples are presented to show our theory. In the first, we analyze the effect of nonoptimal measurement on the estimation precision of the transition frequency for a driven qubit. In the second example, we consider a bipartite system, in which one of them is measurement inaccessible. To be specific, we take a toy model, the nitrogen-vacancy (NV) center in diamond, as the bipartite system, where the NV-center electronic spin interacts with a single nucleus via the dipole-dipole interaction. We achieve a precise estimation for the nuclear Larmor frequency by optimizing only the observables of the electronic spin. In these two examples, the minimum Euclidean distance between an observable and the optimal ones is analyzed and the results show that the closer the observable to the optimal ones the better the estimation precision. This paper establishes a relation between the estimation precision and the distance of the nonoptimal observable to the optimal ones, which could be helpful for experiments seeking the best observable in parameter estimation.
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.
Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology. 1.
Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology.
Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this Review, the authors collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. The authors give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. The authors address the question of how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation.
Abstract The laws of quantum mechanics allow to perform measurements whose precision supersedes results predicted by classical parameter estimation theory. That is, the precision bound imposed by the central limit theorem in the estimation of a broad class of parameters, like atomic frequencies in spectroscopy or external magnetic field in magnetometry, can be overcomewhen using quantum probes. Environmental noise, however, generally alters the ultimate precision that can be achieved in the estimation of an unknown parameter. This tutorial reviews recent theoretical work aimed at obtaining general precision bounds in the presence of an environment.We adopt a complementary approach,wherewe first analyze the problem within the general framework of describing the quantum systems in terms of quantum dynamical maps and then relate this abstract formalism to a microscopic description of the system’s dissipative time evolution.We will show that although some forms of noise do render quantum systems standard quantum limited, precision beyond classical bounds is still possible in the presence of different forms of local environmental fluctuations.