Maximal vectors in Hilbert space and quantum entanglement

One can endow a matrix algebra M with the trace norm to obtain a finite dimensional Banach space L(M). Given two matrix algebras M1, M2, the natural inclusion of L 1(M1⊗M2) in the projective tensor product of Banach spaces L1(M1)⊗̂L 1(M2) is a bijection but not an isometry; and the projective cross norm can be restricted to the convex set S of density matrices in M1 ⊗ M2 to obtain a continuous convex function E : S → [1,∞). We show that E faithfully measures entanglement in the sense that a state is entangled if and only if its density matrix A satisfies E(A) > 1. Moreover, E(A) is maximized at the density matrix A associated with a pure state if and only if the range of A is generated by a maximally entangled unit vector. These concrete results follow from a general analysis of norm-closed subsets V of the unit sphere of a Hilbert space H that are stable under multiplication by complex scalars of absolute value 1. A maximal vector (for V ) is a unit vector ξ ∈ H whose distance to V is maximum d(ξ, V ) = sup ‖η‖=1 d(η, V ), d(ξ, V ) denoting the distance from ξ to the set V . Maximal vectors generalize the maximally entangled unit vectors of quantum theory, since when V is the set of decomposable unit vectors in a tensor product H = H1 ⊗ H2 of two Hilbert spaces, maximal vectors turn out to be exactly the maximally entangled unit vectors. In general, under a mild regularity hypothesis on V we show that there is a norm on L(H) whose restriction to the convex set S of density operators achieves its minimum precisely on the closed convex hull of the rank one projections associated with vectors in V . It achieves its maximum on a rank one projection precisely when its unit vector is a maximal vector. This “entanglement-measuring norm” is unique, and computation shows it to be the projective cross norm in the above setting of bipartite tensor products H = H1 ⊗H2.