Entanglement Witnesses Arising from Exposed Positive Linear Maps

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entanglement can be detected by one of these witnesses, and this witness detects a unique set of entanglement among those. Therefore, they provide a minimal set of witnesses to detect all entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entanglement with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps between three dimensional matrix algebra.