This is an announcement for the paper "Maximal vectors in Hilbert space and quantum entanglement" by William Arveson.
Abstract: Given two matrix algebras $M_1$, $M_2$, the natural inclusion of $\mathcal L^1(M_1\otimes M_2)$ in the projective tensor product of Banach spaces $\mathcal L^1(M_1)\hat\otimes \mathcal L^1(M_2)$ is a contraction but not an isometry; and the projective cross norm can be restricted to the convex set $\mathcal S$ of density matrices in $M_1\otimes M_2$to obtain a continuous convex function $E:\mathcal S\to [1,\infty)$. We show that $E$ {\em 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 $H$. A {\em maximal vector} (for $V$) is a unit vector $\xi\in H$ whose distance to $V$ is maximum. Maximal vectors generalize the ``maximally entangled" unit vectors of quantum theory. In general, under a mild regularity hypothesis on $V$ we show that there is a {\em norm} on $\mathcal L^1(H)$ whose restriction to the convex set $\mathcal S$ 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 $H=H_1\otimes H_2$.
Archive classification: math.OA math.FA
Mathematics Subject Classification: 46N50,81P68, 94B27
Remarks: 25 pages
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Submitted from: arveson@math.berkeley.edu
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