Construction of multi-qubit optimal genuine entanglement witnesses

TL;DR

This paper characterizes and constructs optimal multi-qubit genuine entanglement witnesses, especially for X-shaped matrices, identifying conditions for their optimality and their ability to detect genuine entanglement.

Contribution

It provides a complete characterization of optimal X-shaped entanglement witnesses and links their properties to positivity conditions and spanning properties.

Findings

Identified conditions for X-shaped matrices to be genuine entanglement witnesses

Constructed all optimal X-shaped genuine entanglement witnesses

Demonstrated these witnesses detect a non-zero volume of genuine entanglement

Abstract

We interpret multi-partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them. We apply this result to multi-qubit {\sf X}-shaped Hermitian matrices, and characterize the conditions for them to be genuine entanglement witnesses, in terms of entries. Furthermore, we find all optimal ones among them. They turn out to have the spanning properties, and so they detect non-zero volume set of multi-qubit genuine entanglement. We also characterize decomposability for {\sf X}-shaped entanglement witnesses.