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Generic local distinguishability and completely entangled subspaces

Jon Walgate, Perimeter Institute for Theoretical Physics

(Session 3 : Friday from 15:30-16:00)

Abstract. The geometry of Hilbert space entails many necessary and generic properties of quantum systems. In fact, expressing quantum information theoretic questions in geometric terms can transform apparently complex problems into exceedingly simple results. We present an example - a theorem concerning subspaces of projective Hilbert space with immediate and surprising consequences for entanglement and local state distinguishability.

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, S, approaching the full dimension of the system, D. We show that almost all subspaces with dimension less than or equal to S 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 does not exceed D-S. This condition holds for almost all sets of states of all multipartite systems, and reveals something unexpected. The criterion is identical for separable and for nonseparable states: entanglement makes no difference.

Acknowledgements: Joint work with Andrew Scott, see arXiv:0709.4238