Entanglement witness criteria of strong-k-separability for multipartite quantum states

Let (H_{1}, H_{2},ldots ,H_{n}) be separable complex Hilbert spaces with (dim H_{i}ge 2) and (nge 2). Assume that (rho ) is a state in (H=H_1otimes H_2otimes cdots otimes H_n). (rho ) is called strong-k-separable ((2le kle n)) if (rho ) is separable for any k-partite division of H. In this paper, an entanglement witnesses criterion of strong-k-separability is obtained, which says that (rho ) is not strong-k-separable if and only if there exist a k-division space (H_{m_{1}}otimes cdots otimes H_{m_{k}}) of H, a finite-rank linear elementary operator positive on product states (Lambda :mathcal {B}(H_{m_{2}}otimes cdots otimes H_{m_{k}})rightarrow mathcal {B}(H_{m_{1}})) and a state (rho _{0}in mathcal {S}(H_{m_{1}}otimes H_{m_{1}})), such that (mathrm {Tr}(Wrho )<0), where (W=(mathrm{Id}otimes Lambda ^{dagger })rho _{0}) is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.