Lie Algebraic Uncertainty Relations
Abstract
Uncertainty relations impose fundamental constraints on quantum observables, yet optimal bounds are needed to determine their incompatibility through the minimum attainable uncertainty. Here we introduce a covariance-matrix formulation from which infinitely many uncertainty relations follow, encompassing Hermitian observables and non-Hermitian operators and recovering the variance sum as a special case. Within this family, a basis-independent uncertainty for simple and semisimple Lie algebras admits an exact solution: its optimal state-independent bound is an explicit pairing of root system with the highest weight of the representation. The minimum is attained by a highest-weight state, replacing a search over quantum states with a calculation in representation theory. A complementary result establishes that uncertainty is constant within each weight space, yielding a conservation law. The same commutation relations can therefore support distinct uncertainty limits, depending on their realization on the state space. By rendering these limits analytically accessible, the framework turns representation theory into a tool for quantifying quantum incompatibility and may help establish the boundary between attainable and impossible tasks in quantum theory.
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BibTeX
@misc{hu2026algebraic,
title = {{Lie Algebraic Uncertainty Relations}},
author = {Xiaoli Hu and Naihuan Jing and Jian Zhang and Yunlong Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03661},
eprint = {2610.03661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03661},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lie Algebraic Uncertainty Relations
AU - Xiaoli Hu
AU - Naihuan Jing
AU - Jian Zhang
AU - Yunlong Xiao
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.03661
UR - https://arxiv.org/abs/2610.03661
AB - Uncertainty relations impose fundamental constraints on quantum observables, yet optimal bounds are needed to determine their incompatibility through the minimum attainable uncertainty. Here we introduce a covariance-matrix formulation from which infinitely many uncertainty relations follow, encompassing Hermitian observables and non-Hermitian operators and recovering the variance sum as a special case. Within this family, a basis-independent uncertainty for simple and semisimple Lie algebras admits an exact solution: its optimal state-independent bound is an explicit pairing of root system with the highest weight of the representation. The minimum is attained by a highest-weight state, replacing a search over quantum states with a calculation in representation theory. A complementary result establishes that uncertainty is constant within each weight space, yielding a conservation law. The same commutation relations can therefore support distinct uncertainty limits, depending on their realization on the state space. By rendering these limits analytically accessible, the framework turns representation theory into a tool for quantifying quantum incompatibility and may help establish the boundary between attainable and impossible tasks in quantum theory.
ER -