Partial Commutativity and Quantum Zero-Error Capacity

Recently, it was discovered that there is a connection between the growth factor of the partially commutative monoid, denoted as \(\beta(G)\), and the independence number of the graph. It was shown that \(\lfloor \beta(G) \rfloor \geq \alpha(G)\) and that \(\log \lfloor \beta(G) \rfloor\) is as an upper bound for the classical zero-error capacity~\cite{Silva2023}. In this paper, we demonstrate that \(\beta(G)\) is not only an upper bound for the Lovász number of \(G\), but is also for the chromatic number of the complement \(G\). Furthermore, we show that \(\log \lfloor \beta(G) \rfloor\) is an upper bound for the quantum zero-error capacity.