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.
Categories: quant-ph math.AG