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

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We construct faces of the convex set of all 2⊗ 4 bipartite separable states, which are affinely isomorphic to the simplex Δ₉ 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× 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 Δₖ of Δ₉ share the boundary if and only if k≤ 5. This enables us to find a large class of 2⊗ 4 PPT entangled edge states with rank five.