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

Abstract

Let $H_{1}, H_{2},\ldots ,H_{n}$ be separable complex Hilbert spaces with $\dim H_{i}\ge 2$ and $n\ge 2$. Assume that $\rho$ is a state in $H=H_1\otimes H_2\otimes \cdots \otimes H_n$. $\rho$ is called strong-k-separable $(2\le k\le n)$ 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 $\mathrm {Tr}(W\rho )<0$, 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.