Abstract
A subspace of a multipartite Hilbert space is completely entangled if it contains no product states. Such subspaces can be large with a known maximum size, smax, approaching the full dimension of the system, D. We show that almost all subspaces with dimension s ⩽ smax are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n ⩽ D − smax. This condition holds for almost all sets of states of all multipartite systems and reveals something surprising. The criterion is identical for separable and nonseparable states: entanglement makes no difference.
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