Entropic uncertainty relations under localizations on discrete quantum groups

The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group $\mathbb{G}$ is the free orthogonal quantum group $O_N^+$ with $N\geq 3$ or if $\mathbb{G}$ admits an infinite $\Lambda(p)$ set with $p>2$. On the other hand, this paper explains the reason why such phenomena do not appear when $\mathbb{G}$ is one of the connected semisimple compact Lie groups, $O_2^+$ and the quantum $SU(2)$ groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.