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 $\Delta_{9}$ 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.Comment: 8 pages, 2 figure

Construction of three qubit genuine entanglement with bi-partite positive partial transposes

We construct tri-qubit genuinely entangled states which have positive partial transposes with respect to bi-partition of systems. These examples disprove a conjecture [L. Novo, T. Moroder and O. G\" uhne, Phys.Rev.A {88}, 012305 (2013)] which claims that PPT mixtures are necessary and sufficient for biseparability of three qubits.Comment: 6 page