Additivity of entropic uncertainty relations

René Schwonnek

Institut für Theoretische Physik, Leibniz Universität Hannover, Germany

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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.

For pairs of local measurements performed on multipartite systems, it is shown that the optimal bound in any linear uncertainty relation, formulated in terms of Shannon entropies, is additive. This directly implies that minimal uncertainty can always be realized by fully separable states. Hence, we have an example for a task which is not improved by the use of entanglement. On the other hand, this gives rise to a huge simplification to the structure of global uncertainty bounds as they now can be deduced from local ones by a single letter formula. Furthermore, an extension of the Maassen and Uffink bound for arbitrary linear relations, with positive coefficients, is provided.

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