Year: 2026
Type: preprint
Abstract: 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 $\mathfrak g$. 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 $Δ_ρ^2(\mathfrak g)$ on the multipartite Hilbert space $\mathcal H$ \[ \min_ρΔ_ρ^2(\mathfrak g) = \min_{λ\inΛ(\mathcal H)} 2\langleλ,δ\rangle, \] where $ρ$ is any density operator on $\mathcal H$, $Λ(\mathcal H)$ is the set of highest weights $λ$ labeling those sectors, and $δ$ 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-$1/2$ systems depends only on the parity of the particle number. We further illustrate the framework for multipartite spin-$1$ systems, demonstrating that the same symmetry-sector mechanism persists beyond spin-$1/2$.
Source: arXiv (Cornell University)
Authors: Wang, Yiling, Jing, Naihuan
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204198691