Optimal PPTES witnesses for states in $\mathbb C^n\otimes \mathbb C^n$

TL;DR

This paper demonstrates that certain entanglement witnesses related to indecomposable positive maps are also optimal PPTES witnesses, with their partial transposes exhibiting the spanning property, thus confirming their optimality.

Contribution

It proves that the partial transposes of previously identified optimal entanglement witnesses are also optimal PPTES witnesses, establishing their non-decomposable optimality.

Findings

Partial transposes of these witnesses have spanning property.

These witnesses are confirmed as non-decomposable optimal entanglement witnesses.

The results extend the understanding of optimal PPTES witnesses in quantum entanglement.

Abstract

Recently, X. Qi and J. Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated to indecomposable positive linear maps, but it is not checked whether partial transposes of these witnesses are also optimal. We show that partial transposes of these entanglement witnesses have spanning property, and these witnesses are indeed optimal PPTES witnesses (non-decomposable optimal entanglement witnesses).