Indecomposability of entanglement witnesses constructed from any permutations

Abstract

Let $n\geq 2$ and $\Phi_{n,t,\pi}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})$ be a linear map defined by $\Phi_{n,t,\pi}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,\pi(i)}AE_{i,\pi(i)}^\dag-A$, where $0\leq t\leq n$, $E_{ij}$s are the matrix units and $\pi$ is a non-identity permutation of $(1,2,\cdots,n)$. Denote by $\{{ F}_s: s=1,2\ldots, k\}$ the set of all minimal cycles of $\pi$ and $l(\pi)=\max\{\# { F}_s: s=1,2,\ldots,k\}$ the length of $\pi$. It is shown that the Hermitian matrix $W_{n,t,\pi}$ induced by $\Phi_{n,t,\pi}$ is an indecomposable entanglement witness if and only if $\pi^2\not={\rm id}$ (the identity permutation) and $0<t\leq\frac{n}{l(\pi)}$. Some new bounded entangled states are detected by such witnesses that cannot be distinguished by PPT criterion, realignment criterion, etc..