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arXiv:quant-ph/0509079 · Journal of Physics A Mathematical and General

Construction of 3 otimes 3 entangled edge states with positive partial transposes

Kil-Chan Ha, Seung-Hyeok Kye

Published Sep 28, 2005• 33 citations
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Abstract

We construct a class of 3 ⊗ 3 entangled edge states with positive partial transposes using indecomposable positive linear maps. This class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes.
DOI
10.1088/0305-4470/38/41/014

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We construct faces of the convex set of all $2\ensuremath{\bigotimes}4$ bipartite separable states, which are affinely isomorphic to the simplex ${\ensuremath{\Delta}}_{9}$ with 10 extreme points. Every interior point of these faces is a separable state which has a unique decomposition into 10 product states, even though the ranks of the state and its partial transpose are 5 and 7, respectively. We also note that the number 10 is greater than $2\ifmmode\times\else\texttimes\fi{}4$, to disprove a conjecture on the lengths of qubit-qudit separable states. This face is inscribed in the corresponding face of the convex set of all PPT states so that subsimplices ${\ensuremath{\Delta}}_{k}$ of ${\ensuremath{\Delta}}_{9}$ share the boundary if and only if $k\ensuremath{\le}5$. This enables us to find a large class of $2\ensuremath{\bigotimes}4$ PPT entangled edge states with rank 5.