On the structutre of the algebra generated by the non-commutative operator graph demonstrating superactivation for a zero-error capacity

Abstract

Recently M.E. Shirokov introduced the non-commutative operator graph depending on the complex parameter $\theta $ to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We study the algebraic structure of this graph. The relations for the algebra generated by the graph are derived. In the limiting case $\theta =\pm1$ the graph becomes abelian and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein group.