Bell-state diagonal-entanglement witnesses

Abstract

It has been shown that finding generic Bell states diagonal entanglement witnesses (BDEW) for $d_{1}\otimes d_{2}\otimes ....\otimes d_{n}$ systems exactly reduces to a linear programming if the feasible region be a polygon by itself and approximately obtains via linear programming if the feasible region is not a polygon. Since solving linear programming for generic case is difficult, the multi-qubits, $2\otimes N$ and $3 \otimes 3$ systems for the special case of generic BDEW for some particular choice of their parameters have been considered. In the rest of this paper we obtain the optimal non decomposable entanglement witness for $3 \otimes 3$ system for some particular choice of its parameters. By proving the optimality of the well known reduction map and combining it with the optimal and non-decomposable 3 $\otimes$ 3 BDEW (named critical entanglement witnesses) the family of optimal and non-decomposable 3 $\otimes$ 3 BDEW have also been obtained. Using the approximately critical entanglement witnesses, some 3 $\otimes$ 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 which approximately is obtained via linear programming.