Sharp State-Independent Uncertainty Relations for Multipartite systems
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 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 Δρ2(g) on the multipartite Hilbert space ℋ minρΔρ2(g) = minλ∈Λ(ℋ) 2⟨λ,δ⟩, where ρ is any density operator on ℋ, Λ(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.
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BibTeX
@misc{wang2026sharp,
title = {{Sharp State-Independent Uncertainty Relations for Multipartite systems}},
author = {Yiling Wang and Naihuan Jing},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22656},
eprint = {2608.22656},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22656},
}
RIS
TY - UNPB
TI - Sharp State-Independent Uncertainty Relations for Multipartite systems
AU - Yiling Wang
AU - Naihuan Jing
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.22656
UR - https://arxiv.org/abs/2608.22656
AB - 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. 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(g) on the multipartite Hilbert space ℋ min_ρΔ_ρ^2(g) = min_λ∈Λ(ℋ) 2⟨λ,δ⟩, where ρ is any density operator on ℋ, Λ(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.
ER -