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

Abstract

We study the noncommutative operator graph ${\mathcal L}_{\theta }$ depending on 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 the noncommutative group $G$ and the algebra ${\mathcal A}_{\theta }$ which is a quotient of ${\mathbb C}G$ with respect to the 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 $\theta =\pm 1$ $\phi $ is degenerated to the faithful representation of ${\mathbb C}K_4$, where $K_4$ is the Klein group. Thus, ${\mathcal L}_{\theta }$ can be considered as a noncommutative deformation of the graph associated with the Klein group.