On Asymptotic Quantum Statistical Inference

We study asymptotically optimal statistical inference concerning the unknown state of N identical quantum systems, using two complementary approaches: a "poor man's approach" based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of Le Cam's theory of Local Asymptotic Normality. In the first approach, we make use of a Bayesian version of a quantum Cramer-Rao bound due to Holevo. Holevo's bound can be thought of as a bound on the set of Fisher information matrices for the unknown parameters of the state, as we consider arbitrary measurements on that state. Heuristically one can expect the bound to be asymptotically sharp. We show in various important examples that it is asymptotically attained by measurement-and-estimation schemes which have been proposed by physicists either on ad-hoc grounds or through explicit optimization under rather special prior and loss function. On the way we obtain a family of "dual Holevo bounds" of independent interest. The second approach explains why this all works. The model of N identical unknown quantum states can be approximated locally, at the 1/\sqrt{N} scale, by a single quantum Gaussian model, whose quantum score functions (logarithmic derivatives of the state operator) of one-parametric sub-models have exactly the same inner-products as those of the original (N=1) model, just as in regular parametric i.i.d. models. The convexity relationship between the original Holevo bound and our dual Holevo bound corresponds to the convexity both of the sets of attainable information matrices and of attainable covariance matrices of unbiased estimators in the Gaussian model. The original and the dual Holevo bound are the corresponding lower bound on covariance matrices and the upper bound on information matrices, both attainable in the Gaussian case.