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The information-theoretic formulation of quantum measurement uncertaintyrelations (MURs), based on the notion of relative entropy between measurementprobabilities, is extended to the set of all the spin components for a genericspin s. For an approximate measurement of a spin vector, which givesapproximate joint measurements of the spin components, we define the deviceinformation loss as the maximum loss of information per observable occurring inapproximating the ideal incompatible components with the joint measurement athand. By optimizing on the measuring device, we define the notion of minimuminformation loss. By using these notions, we show how to give a significantformulation of state independent MURs in the case of infinitely many targetobservables. The same construction works as well for finitely many observables,and we study the related MURs for two and three orthogonal spin components. Theminimum information loss plays also the role of measure of incompatibility andin this respect it allows us to compare quantitatively the incompatibility ofvarious sets of spin observables, with different number of involved componentsand different values of s.

Entropic measurement uncertainty relations for all the infinite components of a spin vector