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

We study the noncommutative operator graph ${\mathcal L}_{θ}$ depending on complex parameter $θ$ recently introduced by M.E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define the noncommutative group $G$ and the algebra ${\mathcal A}_{θ}$ which is a quotient of ${\mathbb C}G$ with respect to the special algebraic relation depending on $θ$ such that the matrix representation $ϕ$ of ${\mathcal A}_{θ}$ results in the algebra ${\mathcal M}_{θ}$ generated by ${\mathcal L}_{θ}$. In the case of $θ=\pm 1$ $ϕ$ is degenerated to the faithful representation of ${\mathbb C}K_4$, where $K_4$ is the Klein group. Thus, ${\mathcal L}_{θ}$ can be considered as a noncommutative deformation of the graph associated with the Klein group.

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