LAUSR

Let $$H_{1}, H_{2},\ldots ,H_{n}$$H1,H2,…,Hn be separable complex Hilbert spaces with $$\dim H_{i}\ge 2$$dimHi≥2 and $$n\ge 2$$n≥2. Assume that $$\rho $$ρ is a state in $$H=H_1\otimes H_2\otimes \cdots \otimes H_n$$H=H1⊗H2⊗⋯⊗Hn. $$\rho $$ρ is called strong-k-separable $$(2\le k\le n)$$(2≤k≤n) if $$\rho $$ρ 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 $$\rho $$ρ is not strong-k-separable if and only if there exist a k-division space $$H_{m_{1}}\otimes \cdots \otimes H_{m_{k}}$$Hm1⊗⋯⊗Hmk of H, a finite-rank linear elementary operator positive on product states $$\Lambda :\mathcal {B}(H_{m_{2}}\otimes \cdots \otimes H_{m_{k}})\rightarrow \mathcal {B}(H_{m_{1}})$$Λ:B(Hm2⊗⋯⊗Hmk)→B(Hm1) and a state $$\rho _{0}\in \mathcal {S}(H_{m_{1}}\otimes H_{m_{1}})$$ρ0∈S(Hm1⊗Hm1), such that $$\mathrm {Tr}(W\rho )<0$$Tr(Wρ)<0, where $$W=(\mathrm{Id}\otimes \Lambda ^{\dagger })\rho _{0}$$W=(Id⊗Λ†)ρ0 is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.