The high-dimension limit of characters of compact reductive Lie groups and restrictions on the production of quantum randomness

TL;DR

This paper studies the asymptotic behavior of characters of compact reductive Lie groups in high dimensions, revealing limits that impact quantum randomness production in large quantum systems.

Contribution

It establishes the high-dimension limit of normalized irreducible characters for compact reductive groups, linking representation theory to quantum randomness constraints.

Findings

Normalized characters vanish outside the identity in high dimensions

Results extend to semisimple groups with increasing representation dimensions

Implications for bounds on quantum randomness in large quantum systems

Abstract

For any element $g$ of compact reductive group $G$ we investigate the asymptotic behavior of its normalized irreducible character in the high-dimension limit, $\frac{\chi_\lambda(g)}{d_\lambda}$. We show that when $G$ is simple the limit vanishes besides identity element. For semisimple groups one gets the same results under the additional assumption that dimensions of irreducible representations of all simple components are going to infinity. Using the notion of approximate $t$-designs we connect this observations with bounds on the production of quantum randomness in large quantum systems.