Optimality of a class of entanglement witnesses for $3\otimes 3$ systems

Abstract

Let ${\Phi }_{t,\pi }:M_3(\mathbb{C})\to M_3(\mathbb{C})$ be a linear map defined by ${\Phi }_{t,\pi }(A)=(3-t){\sum }_{i=1}^3{E}_{ii}A{E}_{ii}+t{\sum }_{i=1}^3{E}_{i,\pi (i)}A{E}_{i,\pi (i)}^{†}-A$, where $0\le t\le 3$ and $\pi$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{{\Phi }_{t,\pi }}$ induced by ${\Phi }_{t,\pi }$ is an optimal entanglement witness if and only if $t=1$ and $\pi$ is cyclic.

12 pages

References in corpus (3)

• Detecting Genuine Multipartite Entanglement with Two Local Measurements
• Constructing entanglement witnesses for infinite-dimensional systems
• A characterization of optimal entanglement witnesses