Attaining the ultimate precision limit in quantum state estimation

TL;DR

This paper establishes a fundamental precision limit for quantum state estimation in finite-dimensional systems, demonstrating its attainability under broad conditions using quantum local asymptotic normality.

Contribution

It derives a new bound on quantum estimation precision, proves its attainability in general cases, and extends the results to practical noisy and multi-parameter scenarios.

Findings

Derived a bound on quantum state estimation precision.

Proved the bound is attainable for non-degenerate spectra.

Extended results to noisy and multi-parameter estimation cases.

Abstract

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.