Maximal vectors in Hilbert space and quantum entanglement

Abstract

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 {\em maximal vector} (for $V$) is a unit vector $ξ\in H$ 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 {\em maximally entangled} unit vectors of quantum theory. In general, under a mild regularity hypothesis on $V$, there is a {\em 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 $\mathcal B(H)$ that faithfully detects entanglement of normal states. We apply these abstract results to the analysis of entanglement in multipartite tensor products $H=H_1\otimes ...\otimes H_N$, 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 $H_1\otimes...\otimes H_{N-1}$.