Spectral properties of entanglement witnesses

TL;DR

This paper explores the spectral characteristics of entanglement witnesses, revealing conditions on eigenvectors, restrictions on signatures, and their relation to positive maps within a unified algebraic-geometrical framework.

Contribution

It establishes spectral conditions for entanglement witnesses, links them to positive maps, and generalizes the results within a comprehensive algebraic-geometrical approach.

Findings

Eigenvector relations determine entanglement witness properties

Restrictions on witness signatures are derived from algebraic geometry

Connections between entanglement witnesses and positive maps are established

Abstract

Entanglement witnesses are observables which when measured, detect entanglement in a measured composed system. It is shown what kind of relations between eigenvectors of an observable should be fulfilled, to allow an observable to be an entanglement witness. Some restrictions on the signature of entaglement witnesses, based on an algebraic-geometrical theorem will be given. The set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive, but not completely positive. A translation of the results to the language of positive maps is also given. The properties of entanglement witnesses and positive maps express as special cases of general theorems for $k$-Schmidt witnesses and $k$-positive maps. The results are therefore presented in a general framework.