  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.
