Indecomposable exposed positive bi-linear maps between two by two matrices

TL;DR

This paper constructs specific indecomposable positive bi-linear maps between 2x2 matrices that can detect certain entangled states with positive partial transpose, advancing quantum entanglement detection methods.

Contribution

It introduces explicit indecomposable exposed positive bi-linear maps between 2x2 matrices, which generate extreme rays in the cone of all such maps and detect nonzero volume of PPT entanglement.

Findings

Constructed indecomposable positive bi-linear maps between 2x2 matrices.

These maps generate extreme rays and are exposed, enabling entanglement detection.

They can detect PPT entanglement with nonzero volume.

Abstract

Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between $2\times 2$ matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose whose volume is nonzero.