Almost every set of N≥d+1 orthogonal states on d⊗n is locally indistinguishable

I consider the problem of deterministically distinguishing the state of a multipartite system, from a set of N >= d+1 orthogonal states, where d is the dimension of each party's subsystem. It is shown that if the set of orthogonal states is chosen at random, then there is a vanishing probability that this set will be perfectly distinguishable under the restriction that the parties use only local operations on their subsystems and classical communication among themselves.