  We investigate the additivity properties for both bipartite and multipartite
systems by using entropic uncertainty relations (EUR) defined in terms of the
joint Shannon entropy of probabilities of local measurement outcomes. In
particular, we introduce state-independent and state-dependent entropic
inequalities. Interestingly, the violation of these inequalities is strictly
connected with the presence of quantum correlations. We show that the
additivity of EUR holds only for EUR that involve two observables, while this
is not the case for inequalities that consider more than two observables or the
addition of the von Neumann entropy of a subsystem. We apply them to bipartite
systems and to several classes of states of a three-qubit system.
