Additivity of entropic uncertainty relations

TL;DR

This paper proves that the optimal bounds in entropic uncertainty relations are additive and can be achieved with separable states, simplifying the calculation of global bounds and challenging previous entanglement detection proposals.

Contribution

It establishes the additivity of optimal bounds in entropic uncertainty relations and generalizes the Maassen-Uffink inequality for arbitrary coefficients.

Findings

Optimal bounds are additive in linear entropic uncertainty relations.

Minimal uncertainty states are always separable, not entangled.

Simplifies the computation of global uncertainty bounds from local ones.

Abstract

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.