ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS

• 作者: Jinchuan HOU Wenli WANG
• 来源期刊: 数学物理学报（英文版）
• 年，卷（期）: 2021,（3）
• 页码范围: 843-874
• 页数: 32页
• 语种: 英文
• DOI: 10.3969/j.issn.0252-9602.2021.03.013

摘要

For n≥3,we construct a class{Wn,π1,π2}of n2 x n2 hermitian matrices by the permutation pairs and show that,for a pair{π1,π2}of permutations on(1,2,...,n),Wn,π1,π2 is an entanglement witness of the n?n system if{π1,π2}has the property(C).Recall that a pair{π1,π2}of permutations of(1,2,...,n)has the property(C)if,for each i,one can obtain a permutation of(1,...,i-1,i+1,...,n)from(π1(1),...,π1(i-1),π1(i+1),...,π1(n))and(π2(1),...,π2(i-1),π2(i+1),...,π2(n)).We further prove that Wn,π1,π2 is not comparable with Wn,π,which is the entanglement witness constructed from a single permutation π;Wn,π1,π2 is decomposable if π1π2 = id or π2 = π22 = id.For the low dimensional cases n ∈{3,4},we give a sufficient and necessary condition on π1,π2 for Wn,π1,π2 to be an entanglement witness.We also show that,for n ∈{3,4},Wn,π1,π2 is decomposable if and only if π1π2 = id or π2 = π22 = id;W3,π1,π2is optimal if and only if(π1,π2)=(π,π2),whereπ =(2,3,1).As applications,some entanglement criteria for states and some decomposability criteria for positive maps are established.