Sharp State-Independent Uncertainty Relations for Multipartite systems

Uncertainty relations restrict the fluctuations of incompatible quantum observables. While traditional bounds often depend on the specific quantum state and can become trivial for certain states, state-independent uncertainty relations seek universal bounds that hold across the entire state space. For observables generated by continuous symmetries, sharp state-independent bounds are well established when the underlying symmetry representation is irreducible. However, multipartite collective systems naturally appear as reducible tensor-product representations composed of multiple symmetry sectors. This paper addresses how to determine the total uncertainty in such reducible systems by formulating a representation-theoretic framework.

Using Weyl's complete reducibility theorem, a tensor-product representation of a compact semisimple Lie algebra can be decomposed into isotypic components. The paper proves that the total generator variance for any density operator on a reducible multipartite space admits an exact decomposition into two parts: the weighted sum of intrinsic quantum fluctuations within each irreducible sector, and a nonnegative classical dispersion arising from the separation of their mean generator vectors. This serves as a symmetry-resolved quantum analogue of the law of total variance. Because the total variance splits in this manner, optimizing over the entire quantum state space reduces to minimizing over the discrete set of accessible highest weights labeling the irreducible sectors.

The general framework is illustrated using multipartite spin systems. For collective spin-1/2 systems, the result rigorously confirms a previously conjectured parity effect: odd particle numbers lack a singlet representation and maintain a non-zero uncertainty floor, whereas even particle numbers contain a singlet sector that permits a zero-uncertainty state. Extending beyond spin-1/2, the analysis of multipartite spin-1 systems demonstrates that the same symmetry-sector mechanism dictates the uncertainty floor even when nontrivial multiplicities are present. Ultimately, the representation content of the system, rather than its Hilbert-space dimension, controls the absolute quantum uncertainty.