  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.

Quantum Physics