Entanglement witnesses arising from Choi type positive linear maps

TL;DR

This paper develops new entanglement witnesses based on Choi-type positive maps to detect specific PPT entangled states, offering a method to verify their optimality without relying on the spanning property.

Contribution

It introduces a novel approach to constructing optimal PPT entanglement witnesses using Choi-type maps with complex numbers, and proposes a new method to verify their optimality.

Findings

Constructed optimal PPT entanglement witnesses for specific edge states

Proposed a method to check optimality without the spanning property

Enhanced understanding of entanglement detection techniques

Abstract

We construct optimal PPTES witnesses to detect $3\otimes 3$ PPT entangled edge states of type $(6,8)$ constructed recently \cite{kye_osaka}. To do this, we consider positive linear maps which are variants of the Choi type map involving complex numbers, and examine several notions related to optimality for those entanglement witnesses. Through the discussion, we suggest a method to check the optimality of entanglement witnesses without the spanning property.