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.Comment: 4 pages; minor revision