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The expectation values of operators drawn from a single quantum state cannotbe outside of a particular region, called their allowed region or the jointnumerical range of the operators. Basically, the allowed region is an image ofthe state space under the Born rule. The maximum-eigenvalue-states---of everylinear combination of the operators of interest---are sufficient to generateboundary of the allowed region. In this way, we obtain the numerical range ofcertain Hermitian operators (observables) that are functions of the angularmomentum operators. Especially, we consider here three kinds offunctions---combinations of powers of the ladder operators, powers of theangular momentum operators and their anticommutators---and discover the allowedregions of different shapes. By defining some specific concave (and convex)functions on the joint numerical range, we also achieve tight uncertainty (andcertainty) relations for the observables. Overall, we demonstrate how thenumerical range and uncertainty relations change as the angular momentumquantum number grows. Finally, we apply the quantum de Finetti theorem bytaking a multi-qubit system and attain the allowed regions and tightuncertainty relations in the limit where the quantum number goes to infinity.

Allowed region of the mean values of angular momentum observables and their uncertainty relations