Indecomposability of entanglement witnesses constructed from any permutations

Abstract

Let $n\ge 2$ and ${\Phi }_{n,t,\pi }:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be a linear map defined by ${\Phi }_{n,t,\pi }(A)=(n-t){\sum }_{i=1}^n{E}_{ii}A{E}_{ii}+t{\sum }_{i=1}^n{E}_{i,\pi (i)}A{E}_{i,\pi (i)}^{†}-A$, where $0\le t\le n$, $E_{ij}$s are the matrix units and $\pi$ is a non-identity permutation of $(1,2,\cdots ,n)$. Denote by $\{F_s:s=1,2\dots ,k\}$ the set of all minimal cycles of $\pi$ and $l(\pi )=\max \{\#F_s:s=1,2,\dots ,k\}$ 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 ${\pi }^2\ne \mathrm{id}$ (the identity permutation) and $0<t\le \frac{n}{l(\pi )}$. Some new bounded entangled states are detected by such witnesses that cannot be distinguished by PPT criterion, realignment criterion, etc..

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