  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.
