We present a procedure to share a secret spatial direction in the absence of
a common reference frame using a multipartite quantum state. The procedure
guarantees that the parties 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. As a by-product 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.Comment: 4 pages, 1 figur