Bell-state diagonal-entanglement witnesses

It has been shown that finding generic Bell-state diagonal-entanglement witnesses for d₁⊗d₂⊗⋯⊗d_n systems reduces to linear programming if the feasible region is a polygon by itself, and it can be solved approximately via linear programming if the feasible region is encircled by a polygon. Since solving linear programming for the generic case is difficult, multiqubit, 2⊗N and 3⊗3 systems for the special case of generic Bell-state diagonal-entanglement witnesses for some particular choice of parameters have been considered. We obtain the optimal nondecomposable entanglement witness for a 3⊗3 system for some particular choice of parameters. By proving the optimality of the well-known reduction map and combining it with the optimal and nondecomposable 3⊗3 Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable 3⊗3 Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some 3⊗3 bound entangled states are so detected. So the well-known Choi map as a particular case of the positive map in connection with this witness via Jamiolkowski isomorphism has been considered.