  We consider the uncertainty between two pairs of local projective
measurements performed on a multipartite system. We show that the optimal bound
in any linear uncertainty relation, formulated in terms of the Shannon entropy,
is additive. This directly implies, against naive intuition, that the minimal
entropic uncertainty can always be realized by fully separable states. Hence,
in contradiction to proposals by other authors, no entanglement witness can be
constructed solely by comparing the attainable uncertainties of entangled and
separable states. However, our result gives rise to a huge simplification for
computing global uncertainty bounds as they now can be deduced from local ones.
  Furthermore, we provide the natural generalization of the Maassen and Uffink
inequality for linear uncertainty relations with arbitrary positive
coefficients.
