Matched witness for multipartite entanglement
  We transform the way of finding entanglement criterion into two steps: to obtain necessary criterion of separability by maximizing an algebra function for a set of characteristic variables of the witness operator and the given number of partitions, then to obtain the sufficient criterion by minimizing an algebra function with respect to the characteristic variables for a given quantum state. Our method avoids the semi-definite program calculation in the witness operator entanglement detection. The necessary and sufficient criterion of separability for the three qubit X shaped state is given to illustrate the procedure of finding the criterion. We give the necessary and sufficient criteria of the three partite and full separabilities for the four qubit noisy GHZ state and the four qubit noisy cluster state. 
Xiao-yu Chen, Zhu-an Xu
quant-ph