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

• Published in: Quantum information processing 2018-07, Vol.17 (7), p.1-8, Article 181
• Main Authors: Yan, Siqing, Hou, Jinchuan
• Format: Article
• Language: English
• Subjects: Data Structures and Information Theory; Hilbert space; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum entanglement; Quantum Information Technology; Quantum Physics; Quantum theory; Spintronics
• ISSN: 1570-0755; 1573-1332
• DOI: 10.1007/s11128-018-1952-4

Let H 1 , H 2 , … , H n be separable complex Hilbert spaces with dim H i ≥ 2 and n ≥ 2 . Assume that ρ is a state in H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n . ρ is called strong- k -separable ( 2 ≤ k ≤ n ) if 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 ρ is not strong- k -separable if and only if there exist a k -division space H m 1 ⊗ ⋯ ⊗ H m k of H , a finite-rank linear elementary operator positive on product states Λ : B ( H m 2 ⊗ ⋯ ⊗ H m k ) → B ( H m 1 ) and a state ρ 0 ∈ S ( H m 1 ⊗ H m 1 ) , such that Tr ( W ρ ) < 0 , where W = ( Id ⊗ Λ † ) ρ 0 is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.