Sharp State-Independent Uncertainty Relations for Multipartite systems
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 $\Delta_\rho^2(\mathfrak g)$ on the multipartite Hilbert space $\mathcal H$ \[ \min_{\rho}\Delta_\rho^2(\mathfrak g) = \min_{\lambda\in\Lambda(\mathcal H)} 2\langle\lambda,\delta\rangle, \] where $\rho$ is any density operator on $\mathcal H$, $\Lambda(\mathcal H)$ is the set of highest weights $\lambda$ labeling those sectors, and $\delta$ is the Weyl vector. This demonstrates that the ultimate uncertainty is completely controlled by its intrinsic symmetry structure. As a special example, this result confirms our previous conjecture 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-$1/2$.