Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (3): 843-874.

• Articles •

### ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS

Jinchuan HOU, Wenli WANG

1. School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
• Received:2019-12-24 Revised:2020-09-07 Online:2021-06-25 Published:2021-06-07
• Contact: Jinchuan HOU E-mail:jinchuanhou@aliyun.com
• About author:Wenli WANG,E-mail:995929733@qq.com
• Supported by:
This work is partially supported by National Natural Science Foundation of China (11671294, 12071336).

Abstract: For $n\geq 3$, we construct a class $\{W_{n,\pi_1,\pi_2}\}$ of $n^2\times n^2$ hermitian matrices by the permutation pairs and show that, for a pair $\{\pi_1,\pi_2\}$ of permutations on $(1,2,\ldots,n)$, $W_{n,\pi_1,\pi_2}$ is an entanglement witness of the $n\otimes n$ system if $\{\pi_1,\pi_2\}$ has the property (C). Recall that a pair $\{\pi_1,\pi_2\}$ of permutations of $(1,2,\ldots,n)$ has the property (C) if, for each $i$, one can obtain a permutation of $(1,\ldots,i-1,i+1,\ldots,n)$ from $(\pi_1(1),\ldots,\pi_1(i-1),\pi_1(i+1),\ldots,\pi_1(n))$ and $(\pi_2(1),\ldots,\pi_2(i-1),\pi_2(i+1),\ldots,\pi_2(n))$. We further prove that $W_{n,\pi_1,\pi_2}$ is not comparable with $W_{n,\pi}$, which is the entanglement witness constructed from a single permutation $\pi$; $W_{n,\pi_1,\pi_2}$ is decomposable if $\pi_1\pi_2={\rm id}$ or $\pi_1^2=\pi_2^2={\rm id}$. For the low dimensional cases $n\in\{3,4\}$, we give a sufficient and necessary condition on $\pi_1,\pi_2$ for $W_{n,\pi_1,\pi_2}$ to be an entanglement witness. We also show that, for $n\in\{3,4\}$, $W_{n,\pi_1,\pi_2}$ is decomposable if and only if $\pi_1\pi_2={\rm id}$ or $\pi_1^2=\pi_2^2={\rm id}$; $W_{3,\pi_1,\pi_2}$ is optimal if and only if $(\pi_1,\pi_2)=(\pi,\pi^2)$, where $\pi=(2,3,1)$. As applications, some entanglement criteria for states and some decomposability criteria for positive maps are established.

CLC Number:

• 15B57
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