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

The information-theoretic formulation of quantum measurement uncertainty relations (MURs), based on the notion of relative entropy between measurement probabilities, is extended to the set of all the spin components for a generic spin s. For an approximate measurement of a spin vector, which gives approximate joint measurements of the spin components, we define the device information loss as the maximum loss of information per observable occurring in approximating the ideal incompatible components with the joint measurement at hand. By optimizing on the measuring device, we define the notion of minimum information loss. By using these notions, we show how to give a significant formulation of state independent MURs in the case of infinitely many target observables. The same construction works as well for finitely many observables, and we study the related MURs for two and three orthogonal spin components. The minimum information loss plays also the role of measure of incompatibility and in this respect it allows us to compare quantitatively the incompatibility of various sets of spin observables, with different number of involved components and different values of s.