Four-qubit entangled symmetric states with positive partial transpositions

Abstract

We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical--half-numerical method that allows us to construct multipartite PPT entangled symmetric states (PPTESSs) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing us to search for extremal elements in the convex set of bipartite PPT states [Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESSs and show that generically they have ranks $(5,7,8)$. Finally, we provide an exhaustive characterization of these states with respect to their separability properties.