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

We construct faces of the convex set of all $2\otimes 4$ bipartite separable states, which are affinely isomorphic to the simplex $Δ_{9}$ 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 $Δ_k$ of $Δ_{9}$ 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.

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