Optimality for indecomposable entanglement witnesses

We examine various notions related to the optimality for entanglement witnesses arising from Choi-type positive linear maps. We found examples of optimal entanglement witnesses which are nondecomposable but are not nondecomposable optimal entanglement witnesses in the sense of Lewenstein, Kraus, Cirac, and Horodecki [Phys. Rev. A 62, 052310 (2000)]. We suggest using the terms PPTES witness and optimal PPTES witness in place of nondecomposable entanglement witness and nondecomposable optimal entanglement witnesses in order to avoid possible confusion. Here, PPTES refers entangled states with positive partial transposes. We also found examples of nonextremal optimal entanglement witnesses which are indecomposable.