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arXiv:2610.03661

Lie Algebraic Uncertainty Relations

Xiaoli Hu, Naihuan Jing, Jian Zhang, Yunlong Xiao

quant-phmath.RTSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Oct 2, 2026v2 · Oct 5, 2026
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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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8 pages (main text) + 103 pages (supplemental material). Comments welcome!

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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.
The standard state-dependent Heisenberg-Robertson uncertainty-relation lower bound fails to capture the quintessential incompatibility of observables as the bound can be zero for some states. To remedy this problem, we establish a class of tight (i.e., inequalities are saturated) variance-based sum-uncertainty relations derived from the Lie algebraic properties of observables and show that our lower bounds depend only on the irreducible representation assumed carried by the Hilbert space of state of the system. We illustrate our result for the cases of the Weyl-Heisenberg algebra, special unitary algebras up to rank 4, and any semisimple compact algebra. We also prove the usefulness of our results by extending a known variance-based entanglement detection criterion.
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Abstract Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number j and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated approach, based on the notion of joint numerical range of a number of observables and uncertainty range, allows us to improve earlier numerical works and to derive semianalytical tight bounds for the uncertainty relation for the sum of variances expressed as roots of a polynomial of a single real variable.
State-independent uncertainty relations on multipartite spin 1/21/21/2-systems
Uncertainty relations quantify fundamental limits on simultaneous measurement of quantum observables. While conventional formulations are state-dependent, state-independent uncertainty relations (SIURs) impose universal bounds determined solely by the algebraic structure of the operators, with applications across metrology, quantum cryptography, and entanglement detection. Despite extensive study, exact analytical variance-based SIURs have so far been established primarily for one- and bi-partite systems, while entropic SIURs have been extended to multipartite and memory-assisted settings through information-theoretic constructions. In contrast, exact variance-based SIURs beyond the bipartite level have remained analytically unresolved.} Here we develop a representation-theoretic framework for multipartite SIURs in collective spin-12\tfrac{1}{2}21​ systems. Using the Clebsch--Gordan decomposition and extremal analysis of total spin variance, we derive exact state-independent bounds up to quintipartite systems. A clear structural dichotomy emerges: odd nnn systems exhibit strictly positive universal bounds (e.g., Δ2(su2) ⁣≥ ⁣4/11Δ^2(\mathfrak{su}_2)\!\ge\!4/11Δ2(su2​)≥4/11 for n=3n=3n=3), whereas even nnn admit vanishing variance on trivial sectors but retain positive reduced-space bounds (e.g., Δ2(su2) ⁣≥ ⁣1/8Δ^2(\mathfrak{su}_2)\!\ge\!1/8Δ2(su2​)≥1/8 for n=4n=4n=4). These results establish the first unified, algebraic framework for multipartite variance-based SIURs in qubit ensembles.
The uncertainty principle is fundamentally rooted in the algebraic asymmetry between observables. We introduce a new class of uncertainty relations grounded in the resource theory of asymmetry, where incompatibility is quantified by an observable's intrinsic, state-independent capacity to break the symmetry associated with another. This ``operator asymmetry,'' formalized as the incompatibility norm, leads to a variance-based uncertainty relation for pure states that can be tighter than the standard Robertson bound. Most significantly, this framework resolves a long-standing open problem in quantum information theory: the formulation of a universally valid, product-form uncertainty relation for the Wigner-Yanase skew information. We demonstrate the practical power of our framework by deriving tighter quantum speed limits for the dynamics of nearly conserved quantities, which are crucial for understanding non-equilibrium phenomena such as prethermalization and many-body localization. This work provides both a new conceptual lens for understanding quantum uncertainty and a powerful, versatile toolkit for its application.
Uncertainty relations constrain the fluctuations of incompatible observables, but most familiar bounds depend on the quantum state. State-independent uncertainty relations instead ask how much fluctuation remains unavoidable for every quantum state. For observables generated by a continuous symmetry, sharp state-independent bounds are known when the symmetry representation is irreducible. Multipartite collective systems, however, generally appear as reducible tensor-product representations of a symmetry algebra, which raises the question of how to determine their total uncertainty. We resolve this problem for multipartite quantum systems with a compact semisimple symmetry algebra g\mathfrak gg. Exploiting the symmetry structure, we formulate a general framework for state-independent uncertainty based on representation theory. The total variance admits an exact decomposition into intrinsic fluctuations within irreducible sectors and a nonnegative dispersion between sectors. This yields the sharp state-independent bound for the total variance Δρ2(g)Δ_ρ^2(\mathfrak g)Δρ2​(g) on the multipartite Hilbert space H\mathcal HH min⁡ρΔρ2(g)=min⁡λ∈Λ(H)2⟨λ,δ⟩, \min_ρΔ_ρ^2(\mathfrak g) = \min_{λ\inΛ(\mathcal H)} 2\langleλ,δ\rangle, minρ​Δρ2​(g)=minλ∈Λ(H)​2⟨λ,δ⟩, where ρρρ is any density operator on H\mathcal HH, Λ(H)Λ(\mathcal H)Λ(H) is the set of highest weights λλλ labeling those sectors, and δδδ is the Weyl vector. This demonstrates that the ultimate uncertainty is completely controlled by its intrinsic symmetry structure. As an application of the general theory, we confirm our previous conjecture [JPA 59, 155302 (2026)] that the total uncertainty floor of collective spin-1/21/21/2 systems depends only on the parity of the particle number. We further illustrate the framework for multipartite spin-111 systems, demonstrating that the same symmetry-sector mechanism persists beyond spin-1/21/21/2.