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

Xiaofei Qi, Jinchuan Hou

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…