Estimation of pure qubit states with collective and individual measurements

We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is formally equivalent to phase estimation; the second one, where there is no constrain on the state, can also be regarded as the estimation of a direction in space using a quantum arrow made out of $N$ parallel spins. We discuss various schemes with and without classical communication and compare their efficiency. We show that the fidelity of the most general collective measurement can always be achieved asymptotically with local measurements and no classical communication.