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 $\varphi$ of ${\mathcal{A}}_{\theta }$ results in the algebra ${\mathcal{M}}_{\theta }$ generated by ${\mathcal{L}}_{\theta }$. In the case of $\theta =\pm 1$ $\varphi$ 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.

22 pages, extended version of arXiv:1512.02096