Mathematics – Operator Algebras
Scientific paper
2008-04-08
Mathematics
Operator Algebras
Significant clarification of results on multipartite tensor products
Scientific paper
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 $\xi\in H$ whose distance to $V$ is maximum $d(\xi,V)=\sup_{\|\eta\|=1}d(\eta,V)$, $d(\xi,V)$ denoting the distance from $\xi$ 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 $H_1\otimes...\otimes H_{N-1}$.
No associations
LandOfFree
Maximal vectors in Hilbert space and quantum entanglement does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Maximal vectors in Hilbert space and quantum entanglement, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Maximal vectors in Hilbert space and quantum entanglement will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-539675