State estimation for large ensembles

We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant measure of the quality of estimation becomes the mean quadratic error matrix. Here we present a bound on this quantity: a quantum Cram\'er-Rao inequality. This bound succinctly expresses how in the quantum case one can trade information about one parameter for information about another. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states---a striking example of nonlocality without entanglement for mixed but not for pure states. Cram\'er-Rao bounds are generally only derived for unbiased estimators. Here we give a version of our bound for biased estimators, and a simple asymptotic version for large N. Finally we prove that when the unknown state belongs to a two-dimensional Hilbert space our quantum Cram\'er-Rao bound can always be attained, and we provide an explicit measurement strategy that attains it. Thus we have a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. The same is true for qubits in the same mixed state if one restricts oneself to separable measurements, but nonseparable measurements allow dramatic increase of efficiency. Exactly how much increase is possible is a major open problem.