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ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS
Jinchuan HOU, Wenli WANG
Acta mathematica scientia,Series B. 2021, 41 (3 ):
843-874.
DOI: 10.1007/s10473-021-0313-z
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.
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