Communication of spin directions with product states and finite measurements

Total spin eigenstates can be used to intrinsically encode a direction, which can later be decoded by means of a quantum measurement. We study the optimal strategy that can be adopted if, as is likely in practical applications, only product states of N spins are available. We obtain the asymptotic behavior of the average fidelity, which provides a proof that the optimal states must be entangled. We also give a prescription for constructing finite measurements for general encoding eigenstates.