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Journal of Mathematical Physics

Asymptotic performance of optimal state estimation in qubit system

Masahito Hayashi, Keiji Matsumoto

Published Oct 1, 2008• 101 citations
Publisher

Abstract

We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.

DOI
10.1063/1.2988130

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We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.
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.
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result. Often one has a choice of what measurement to take, corresponding to different experimental set-ups or settings of measurement apparatus. This leads to a design problem--which measurement is best for a given statistical problem. This paper gives an introduction to this field in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle. We show how in some cases asymptotically optimal measurements can be constructed. Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive. In physical terms, we have non-locality without entanglement.
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random.If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result.Often one has a choice of what measurement to take, corresponding to different experimental set-ups or settings of measurement apparatus.This leads to a design problem-which measurement is best for a given statistical problem.This paper gives an introduction to this field in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle.We show how in some cases asymptotically optimal measurements can be constructed.Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive.In physical terms, we have non-locality without entanglement.
We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant measure of the quality of estimation becomes the mean quadratic error matrix. Here we present a bound on this quantity: a quantum Cram\'er-Rao inequality. This bound succinctly expresses how in the quantum case one can trade information about one parameter for information about another. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states---a striking example of nonlocality without entanglement for mixed but not for pure states. Cram\'er-Rao bounds are generally only derived for unbiased estimators. Here we give a version of our bound for biased estimators, and a simple asymptotic version for large N. Finally we prove that when the unknown state belongs to a two-dimensional Hilbert space our quantum Cram\'er-Rao bound can always be attained, and we provide an explicit measurement strategy that attains it. Thus we have a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. The same is true for qubits in the same mixed state if one restricts oneself to separable measurements, but nonseparable measurements allow dramatic increase of efficiency. Exactly how much increase is possible is a major open problem.
We discuss a problem of parameter estimation for quantum two-level system, qubit system, in presence of unknown phase parameter. We analyze trade-off relations for mean-square errors when estimating relevant parameters with separable measurements based on known precision bounds; the symmetric logarithmic derivative Cramer-Rao bound and Hayashi-Gill-Massar (HGM) bound. We investigate the optimal measurement which attains the HGM bound and discuss its properties. We show that the HGM bound for relevant parameters can be attained asymptotically by using some fraction of given nnn quantum states to estimate the phase parameter. We also discuss the Holevo bound which can be attained asymptotically by a collective measurement.