State estimation for large ensembles

Abstract

We consider the problem of estimating the state of a large but nite number N of identical quantum systems In the limit of large N the problem simplies In particular the only relevant measure of the quality of the estimation is the mean quadratic error matrix Here we present a bound on the mean quadratic error which is a new quantum version of the CramerRao inequality This new bound expresses in a succinct way how in the quantum case one can trade information about one parameter for information about another parameter The bound holds for arbitrary measurements on pure states but only for separable measurements on mixed statesa striking example of non-locality without entanglement for mixed but not for pure states CramerRao bounds are generally derived under the assumption that the estimator is unbiased We also prove that under additional regularity conditions our bound also holds for biased estimators Finally we prove that when the unknown states belong to a dimensional Hilbert space our quantum CramerRao bound can always be attained and we provide an explicit measurement strategy that attains our bound This therefore provides a complete solution to the problem of estimating as e ciently as possible the unknown state of a large ensemble of qubits in the same pure state For qubits in the same mixed state this also provides an optimal estimation strategy if one only considers separable measurements