Indecomposability of entanglement witnesses constructed from any permutations

TL;DR

This paper investigates the indecomposability of entanglement witnesses derived from permutation-based linear maps, identifying conditions under which they detect certain bound entangled states beyond standard criteria.

Contribution

It provides a characterization of when permutation-based entanglement witnesses are indecomposable, expanding the tools for detecting bound entangled states.

Findings

Indecomposability occurs if and only if the permutation squared is not the identity and a parameter condition is met.

New bound entangled states are detected that are not identified by PPT or realignment criteria.

The paper links permutation cycle structure to entanglement witness properties.

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..