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

Piotr Borodako, Adam Sawicki

TL;DR

This work analyzes the high-dimension asymptotics of normalized irreducible characters χ_λ(g)/d_λ on compact reductive groups, proving vanishing for all non-identity elements when G is simple, and extending to semisimple groups provided the simple factors' dimensions grow unbounded. The authors deploy the Weyl Character Formula, a detailed singular-element analysis, and a divergence lemma rooted in Dynkin diagram connectivity to derive a sharp criterion (Theorem cor:critdiv) for when the limit is zero. Geometrically, the results tie the asymptotics to the stabilizer and the centralizer of the singular element, revealing a universal spectral behavior that reduces to the Kesten–McKay law for averaging operators on large irreps. Practically, the findings impose a universal speed limit for producing approximate unitary t-designs via random walks, independent of the ambient symmetry group, with direct implications for quantum randomness generation and the efficiency limits of pseudo-random unitaries.

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, $\frac{χ_λ(g)}{d_λ}$. 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.

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

TL;DR

This work analyzes the high-dimension asymptotics of normalized irreducible characters χ_λ(g)/d_λ on compact reductive groups, proving vanishing for all non-identity elements when G is simple, and extending to semisimple groups provided the simple factors' dimensions grow unbounded. The authors deploy the Weyl Character Formula, a detailed singular-element analysis, and a divergence lemma rooted in Dynkin diagram connectivity to derive a sharp criterion (Theorem cor:critdiv) for when the limit is zero. Geometrically, the results tie the asymptotics to the stabilizer and the centralizer of the singular element, revealing a universal spectral behavior that reduces to the Kesten–McKay law for averaging operators on large irreps. Practically, the findings impose a universal speed limit for producing approximate unitary t-designs via random walks, independent of the ambient symmetry group, with direct implications for quantum randomness generation and the efficiency limits of pseudo-random unitaries.

Abstract

For any element of compact reductive group we investigate the asymptotic behavior of its normalized irreducible character in the high-dimension limit, . We show that when 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 -designs we connect this observations with bounds on the production of quantum randomness in large quantum systems.

Paper Structure

This paper contains 16 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction and motivations
  2. Theoretical Framework: The Weyl Character Formula
  3. A Motivating Example: The SU$(2)$ Limit
  4. Asymptotic Analysis at Singular Elements
  5. Case Study: The SU$(3$) Character at a Singular Point
  6. The General Case: SU$(N)$ Characters
  7. Discussion and Implications
  8. Interpretation: The Stabilizer and the Centralizer
  9. Generalizations and Limitations
  10. Classical Simple Algebras
  11. Type $B_n$: The $\mathfrak{so}(2n+1)$ Algebra
  12. Type $C_n$: The $\mathfrak{sp}(n)$ Algebra
  13. Exceptional Simple Algebras
  14. Non-Simple Reductive Algebras
  15. Application to spectral measure of averaging operators in large irreps
  16. ...and 1 more sections

Key Result

Theorem 4.1

For any fixed singular element $g \neq I$ in SU$(3)$, corresponding to a Cartan element $h_0 \neq 0$, the normalized character vanishes in the high-dimension limit:

Theorems & Definitions (22)

  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 12 more