>VITAE:
Richard Gill is Professor in Mathematical Statistics at the University of Utrecht, Netherlands. He is a member of the Dutch Academy of Sciences. In recent years he became involved in Quantum Mechanics, and has been working on the statistical approach to quantum state reconstruction problems, and on statistical aspects of Bell-type experiments.

Suppose we are given N copies of a state rho depending on a vector of unknown parameters theta. Suppose we also have a prior distribution for theta and a smooth loss function (e.g., 1-fidelity), and we want to choose a (collective) measurement and accompanying data-processing (an estimator) in order to minimize the expected loss. I conjecture that under mild conditions on the prior and on the family of states, the following is a sharp asymptotic lower bound to N times the risk: the average with respect to the prior distribution of the Holevo quantum Cramer-Rao bound C computed for the local quadratic approximation to the loss function.
I will prove the result in some important special cases using the van Trees inequality, and show how the result would be true under a very plausible relative compactness conjecture concerning sequences of quantum measurements on product systems, which seems worth investigating.