Optimality for indecomposable entanglement witnesses

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 termPPTES witness' and optimal PPTES witness' in the places ofnon-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.