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

TL;DR

This paper investigates the algebraic structure of a non-commutative operator graph related to quantum channels, revealing its properties and degenerations, which are relevant for understanding zero-error quantum capacities.

Contribution

It derives the relations for the algebra generated by the operator graph and analyzes its structure, especially in the degenerate case where it becomes abelian.

Findings

The algebra relations for the non-commutative operator graph are explicitly derived.

In the limit $ heta = ext{±}1$, the graph simplifies to an abelian algebra.

The degenerate case corresponds to a direct sum of four one-dimensional irreducible representations of the Klein group.

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.