Collective versus local measurements on two parallel or antiparallel spins

We give a complete analysis of covariant measurements on two spins. We consider the cases of two parallel and two antiparallel spins, and we consider both collective measurements on the two spins, and measurements which require only Local Quantum Operations and Classical Communication (LOCC). In all cases we obtain the optimal measurements for arbitrary fidelities. In particular we show that if the aim is determine as well as possible the direction in which the spins are pointing, it is best to carry out measurements on antiparallel spins (as already shown by Gisin and Popescu), second best to carry out measurements on parallel spins and worst to be restricted to LOCC measurements. If the the aim is to determine as well as possible a direction orthogonal to that in which the spins are pointing, it is best to carry out measurements on parallel spins, whereas measurements on antiparallel spins and LOCC measurements are both less good but equivalent.