Amount of information obtained by a quantum measurement

In this paper we address the problem of how much can be learned about an unknown quantum state by a measurement. To this end we consider optimal measurements for the state estimation problem, that is measurements that maximize the expectation of a fidelity function. We then enlarge the class of optimal measurements to measurements that act collectively on blocks of input states, 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 optimal measurements described above one can always construct an optimal measurement so that the Shannon information of its outputs equals the von Neumann entropy of the unknown states. Since this result is valid for all choices of fidelity functions and all distributions of input states, it provides a model independent answer to the question of how much can be learned about a quantum state by a measurement. Namely, this result shows that a measurement can extract at most one meaningful bit from every qubit carried by the unknown state.