Amount of information obtained by a quantum measurement - art. no. 062303

In this paper we address the problem of how much can be learned about an un known quantum state by a measurement. To this end we consider optimal measu rements for the state estimation problem, that is measurements that maximiz e the expectation of a fidelity function. We then enlarge the class of opti mal measurements to measurements that act collectively on blocks of input s tates, and in addition we only require that the fidelity of the measurement be arbitrarily close to the optimal fidelity. We then consider the Shannon information of the outputs of optimal measurements, which is the amount of data produced by the measurements. We show that in the enlarged class of o ptimal measurements described above one can always construct an optimal mea surement so that the Shannon information of its outputs equals the von Neum ann entropy of the unknown states. Since this result is valid for all choic es of fidelity functions and all distributions of input states, it provides a model independent answer to the question of how much can be learned abou t a quantum state by a measurement. Namely, this result shows that a measur ement can extract at most one meaningful bit from every qubit carried by th e unknown state.