On the noncommutative deformation of the operator graph corresponding to the Klein group

We study the noncommutative operator graph $\mathcal L_\theta$ depending on a complex parameter $\theta$ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing $n$-shot capacity. We define a noncommutative group $G$ and an algebra $\mathcal A_\theta$ which is a quotient of $\mathbb CG$ with respect to a special algebraic relation depending on $\theta$ such that the matrix representation $\phi$ of $\mathcal A_\theta$ results in the algebra $\mathcal M_\theta$ generated by $\mathcal L_\theta$. In the case of $\theta=\pm1$, the representation $\phi$ degenerates into an faithful representation of $\mathbb CK_4$, where $K_4$ is the Klein group. Thus, $\mathcal L_\theta$ can be regarded as a noncommutative deformation of the graph associated with the Klein group.