How to hide a secret direction

We present a procedure that uses a multipartite quantum state to communicate a secret spatial direction when there is no shared reference frame between the preparer and the group of recipients. The procedure guarantees that the recipients can determine the direction if they perform joint measurements on the state, but fail to do so if they restrict themselves to local operations and classical communication (LOCC). We calculate the fidelity for joint measurements, give bounds on the fidelity achievable by LOCC, and prove that there is a non-vanishing gap between the two of them, even in the limit of infinitely many copies. The robustness of the procedure under particle loss is also studied. Additionally, we find bounds on the probability of discriminating by LOCC between the invariant subspaces of total angular momentum N/2 and N/2 - 1 in a system of N elementary spins. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.