Manipulating multiqudit entanglement witnesses by using linear programming

A class of entanglement witnesses (EWs) called reduction-type entanglement witnesses is introduced, which can detect some multipartite entangled states including positive partial transpose ones with Hilbert space of dimension d₁⊗d₂⊗⋯⊗d_n. In fact the feasible regions of these EWs turn out to be convex polygons and hence the manipulation of them reduces to linear programming which can be solved exactly by using the simplex method. The decomposability and nondecomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamiołkowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Hall Phys. Rev. A 72 022311 (2005)] are obtained.