Optimality for indecomposable entanglement witnesses

TL;DR

This paper investigates the properties of optimal entanglement witnesses derived from Choi type positive maps, highlighting distinctions between decomposable and non-decomposable cases and proposing clearer terminology.

Contribution

It introduces new examples of optimal entanglement witnesses, clarifies their classification, and proposes terminology to avoid confusion in the field.

Findings

Found optimal entanglement witnesses that are non-decomposable but not 'non-decomposable optimal'

Identified non-extremal optimal entanglement witnesses that are indecomposable

Suggested terminology changes to improve clarity in the literature

Abstract

We examine various notions related with the optimality for entanglement witnesses arising from Choi type positive linear maps. We found examples of optimal entanglement witnesses which are non-decomposable, but which are not `non-decomposable optimal entanglement witnesses' in the sense of [M. Lewenstein, B. Kraus, J. Cirac, and P. Horodecki, Phys. Rev. A 62, 052310 (2000)]. We suggest to use the term `PPTES witness' and `optimal PPTES witness' in the places of `non-decomposable entanglement witness' and `non-decomposable optimal entanglement witnesses' in order to avoid possible confusion. We also found examples of non-extremal optimal entanglement witnesses which are indecomposable.