The asymptotics of the optimal covariant estimate of a shift parameter of a pure quantum state is studied. In the regular case, where the variance of the generator of the shift group is finite, this problem of quantum statistics turns outto be connected with the information-theoretic approach to the classical central limit theorem proposed by Linnik in 1959. We also consider the quantum analogue of estimation of the shift parameter of the uniform distribution in which the regularity assumption is violated. The advantage of using entanglement is demonstrated in the class of covariant estimates.
The asymptotics of the optimal covariant estimate of a shift parameter of a pure quantum state is studied. In the regular case, where the variance of the generator of the shift group is finite, this problem of quantum statistics turns outto be connected with the information-theoretic approach to the classical central limit theorem proposed by Linnik in 1959. We also consider the quantum analogue of estimation of the shift parameter of the uniform distribution in which the regularity assumption is violated. The advantage of using entanglement is demonstrated in the class of covariant estimates.
We develop an asymptotic theory of estimation of a shift parameter in a pure quantum state to study the relation between entangled and unentangled covariant estimates in the analytically most transparent way. After recollecting basics of estimation of shift parameter in Sec. 2, we study the structure of the optimal covariant estimate in Sec. 3, showing how entanglement comes into play for several independent trials. In Secs. 4,5 we give the asymptotics of the performance of the optimal covariant estimate comparing it with the ``semiclassical'' unentangled covariant estimation in the regular case of finite variance of the generator of the shift group. Sec. 6 is devoted to covariant estimation in the case where the regularity assumption is violated.
The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number N of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies (Guţă M 2011 Phys. Rev. A 83 062324 ; van Horssen M and Guţă M 2015 J. Math. Phys. 56 022109 ) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.
In classical statistics, Pinsker’s theorem provides an exact asymptotic minimax bound in nonparametric estimation, improving upon optimal rates of convergence results. We obtain a quantum version of the theorem by establishing asymptotic minimax results for estimation of the displacement vector in a quantum Gaussian white noise model, given by a sequence of shifted vacuum states. Analogous results are then obtained for estimation of a general pure state from an ensemble of identically prepared, independent quantum systems, using the recently established local asymptotic equivalence to the quantum Gaussian white noise model. Optimality holds with respect to the most fundamental distance measure for quantum states, that is, trace norm distance, and in a true quantum sense, allowing for all possible measurements. Adaptive estimators are also obtained for the above cases. As an application, we obtain asymptotic minimax adaptive estimators for Wigner functions of pure states.
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.