Optimal indecomposable witnesses without extremality as well as spanning property

TL;DR

This paper investigates the existence of certain optimal indecomposable entanglement witnesses and concludes negatively, showing that such witnesses must have either the spanning property or be associated with extremal positive linear maps.

Contribution

The paper proves that no optimal indecomposable entanglement witness exists without the spanning property or extremality, resolving a key open question in quantum entanglement theory.

Findings

No optimal indecomposable witness without spanning property or extremality exists.

Extremality of positive linear maps constructed by Qi and Hou is crucial.

The negative result clarifies the structure of optimal entanglement witnesses.

Abstract

One of the interesting problems on optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable witness which neither has the spanning property nor is associated with extremal positive linear map. Here, we answer this question negatively by examining the extremality of the positive linear maps constructed by Qi and Hou [J. Phys. A {\bf 44}, 215305 (2100)].