Geometry for separable states and construction of entangled states with positive partial transposes

Abstract

We construct faces of the convex set of all $2\otimes 4$ bipartite separable states, which are affinely isomorphic to the simplex ${\Delta }_9$ with ten extreme points. Every interior point of these faces is a separable state which has a unique decomposition into 10 product states, even though ranks of the state and its partial transpose are 5 and 7, respectively. We also note that the number 10 is greater than $2\times 4$, 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 sub-simplices ${\Delta }_k$ of ${\Delta }_9$ share the boundary if and only if $k\le 5$. This enables us to find a large class of $2\otimes 4$ PPT entangled edge states with rank five.

8 pages, 2 figures

References in corpus (4)

• Qubit-qudit states with positive partial transpose
• Separable states with unique decompositions
• Dimensions, lengths and separability in finite-dimensional quantum systems
• Faces for two qubit separable states and the convex hulls of trigonometric moment curves

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• Separability criterion for three-qubit states with a four dimensional norm
• Length filtration of the separable states
• Constructing a ball of separable and absolutely separable states for $2\otimes d$ quantum system
• Three qubit separable states of length ten with unique decompositions
• The number of product vectors and their partial conjugates in a pair of spaces
• Global geometric difference between separable and Positive partial transpose states