Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction
Abstract
Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles---including bosons and symmetric collections of spin-1/2 particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of suℝ(2), contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-n/2 representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto suℝ(n+1) and uℝ(n+1), establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.
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BibTeX
@misc{heib2026universality,
title = {{Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction}},
author = {Tim Heib and Pérola Milman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15302},
eprint = {2609.15302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15302},
}
RIS
TY - UNPB
TI - Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction
AU - Tim Heib
AU - Pérola Milman
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15302
UR - https://arxiv.org/abs/2609.15302
AB - Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles---including bosons and symmetric collections of spin-1/2 particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of su_ℝ(2), contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-n/2 representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto su_ℝ(n+1) and u_ℝ(n+1), establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.
ER -