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.
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The authors wish to give their thanks to the referees for many helpful comments and suggestions to improve original manuscript.
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This work is partially supported by National Natural Science Foundation of China (No. 11671294).
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Yan, S., Hou, J. Entanglement witness criteria of strong-k-separability for multipartite quantum states. Quantum Inf Process 17, 181 (2018). https://doi.org/10.1007/s11128-018-1952-4
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DOI: https://doi.org/10.1007/s11128-018-1952-4