Entropic uncertainty relations under localizations on discrete quantum groups

TL;DR

This paper explores how entropic uncertainty relations can be enhanced under localizations on discrete quantum groups, revealing specific conditions where these improvements occur and explaining cases where they do not.

Contribution

It demonstrates the strengthening of entropic uncertainty relations under localizations on certain discrete quantum groups and clarifies why this does not happen for some classical and quantum groups.

Findings

Strengthening of entropic uncertainty relations for free orthogonal quantum groups $O_N^+$ with $N extgreater 3$.

Enhancement of relations when the dual quantum group admits an infinite $ ext{Lambda}(p)$ set with $p>2$.

Explanation of why such phenomena are absent in connected semisimple compact Lie groups, $O_2^+$, and quantum $SU(2)$ groups.

Abstract

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.