Maximal vectors in Hilbert space and quantum entanglement

Let V be a norm-closed subset of the unit sphere of a Hilbert space H that is stable under multiplication by scalars of absolute value 1. A maximal vector (for V) is a unit vector xi is an element of H whose distance to V is maximum d(xi, V) = sup d(eta, v), parallel to eta parallel to = 1 d(xi, V) denoting the distance from xi to the set V. Maximal vectors generalize the maximally entangled unit vectors of quantum theory. In general, under a mild regularity hypothesis on V, there is a norm on H whose restriction to the unit sphere achieves its minimum precisely on V and its maximum precisely on the set of maximal vectors. This "entanglement-measuring norm" is unique. There is a corresponding "entanglement-measuring norm" on the predual of B(H) that faithfully detects entanglement of normal states. We apply these abstract results to the analysis of entanglement in multipartite tensor products H = H(1) circle times ... circle times H(N), and we calculate both entanglement-measuring norms. In cases for which dim H(N) is relatively large with respect to the others, we describe the set of maximal vectors in explicit terms and show that it does not depend on the number of factors of the Hilbert space H(1) circle times ... circle times H(N-1). (C) 2008 Elsevier Inc. All rights reserved.