Separability of qubit-qudit quantum states with strong positive partial transposes

TL;DR

This paper proves all 2x4 states with strong positive partial transposes are separable, constructs entangled SPPT states for 2x5, and clarifies the relation between separable and SPPT states for 2xd systems.

Contribution

It settles the conjecture on the separability of SPPT states by proving all 2x4 SPPT states are separable and providing counterexamples for 2x5.

Findings

All 2x4 SPPT states are separable.

Existence of entangled 2x5 SPPT states.

Clarification of the relation between separable and SPPT states for 2x3 and 2x4 systems.

Abstract

We show that all $2\otimes 4$ states with strong positive partial transposes (SPPT) are separable. We also construct a family of $2\otimes 5$ entangled SPPT states, so the conjecture on the separability of SPPT states are completely settled. In addition, we clarify the relation between the set of all $2\otimes d$ separable states and the set of all $2\otimes d$ SPPT states for the case of $d=3,4$.