Four-qubit entangled symmetric states with positive partial transpositions

TL;DR

This paper demonstrates the existence of four-qubit entangled symmetric states with positive partial transpositions using novel analytical and numerical methods, and characterizes their separability properties.

Contribution

It introduces new methods to construct and analyze four-qubit PPT entangled symmetric states, solving a previously open problem.

Findings

Existence of four-qubit PPT entangled symmetric states confirmed.

Most extremal states have ranks (5,7,8).

Complete characterization of these states' separability properties.

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 to construct multipartite PPT entangled symmetric states (PPTESS) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing to search for extremal elements in the convex set of bipartite PPT states [J. M. Leinaas, J. Myrheim, and E. Ovrum, Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESS 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.