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

TL;DR

This paper constructs specific faces of the set of bipartite separable states in 2x4 systems, revealing unique decompositions and providing new examples of PPT entangled states, thereby disproving a conjecture on separable state lengths.

Contribution

It introduces new faces of separable states with unique decompositions and constructs PPT entangled edge states, challenging existing conjectures on separable state lengths.

Findings

Faces of separable states with unique decompositions are constructed.

Disproves conjecture on lengths of qubit-qudit separable states.

Identifies a large class of PPT entangled edge states with rank five.

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.