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 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 $2\times 4$, 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 subsimplices ${\Delta }_k$ 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 5.

• Received 13 July 2013

DOI:https://doi.org/10.1103/PhysRevA.88.024302