Bell-state diagonal-entanglement witnesses
M. A. Jafarizadeh, M. Rezaee, S. K. A. Seyed Yagoobi · Physical Review A · 2005
It has been shown that finding generic Bell-state diagonal-entanglement witnesses for ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ 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, $2\ensuremath{\bigotimes}N$ and $3\ensuremath{\bigotimes}3$ 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 $3\ensuremath{\bigotimes}3$ 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 $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some $3\ensuremath{\bigotimes}3$ 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.