Device-independent uncertainty for binary observables

We investigate entropic uncertainty relations for two or more binary measurements. We show that the effective anti-commutator is a useful measure of incompatibility and gives rise to strong uncertainty relations. Since the effective anti-commutator can be certified device-independently it leads us to device-independent uncertainty. Our relations, expressed in terms of conditional Rényi entropies, turn out to be robust (they give non-trivial bounds on the uncertainty whenever deterministic behaviour cannot be ruled out) and strong (e.g. for the well-studied case of two projective measurements on a qubit we find an analytic expression that improves on the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] and some recent bounds based on the majorisation approach). In addition to being a useful tool towards robust, device-independent quantum cryptography beyond quantum key distribution (QKD), our results are also interesting from the technical point of view since the methods used rely solely on standard matrix analysis and differ substantially from the techniques usually employed in deriving uncertainty relations.