We improve the entropic uncertainty relations for position and momentum
coarse-grained measurements. We derive the continuous, coarse-grained
counterparts of the discrete uncertainty relations based on the concept of
majorization. The obtained entropic inequalities involve two R\'enyi entropies
of the same order, and thus go beyond the standard scenario with conjugated
parameters. In a special case describing the sum of two Shannon entropies the
majorization-based bounds significantly outperform the currently known results
in the regime of larger coarse graining, and might thus be useful for
entanglement detection in continuous variables.Comment: 6 pages, final versio
