Tight Bounds from Multiple-Observable Entropic Uncertainty Relations

The additivity properties for both bipartite and multipartite systems are investigated by using entropic uncertainty relations (EUR) defined in terms of the joint Shannon entropy of probabilities of local measurement outcomes. In particular, state-independent and state-dependent entropic inequalities are introduced. Interestingly, the violation of these inequalities is strictly connected with the presence of quantum correlations. It is shown 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. They are applied to bipartite systems and to several classes of states of a three-qubit system.