  Uncertainty relations give upper bounds on the accuracy by which the outcomes
of two incompatible measurements can be predicted. While established
uncertainty relations apply to cases where the predictions are based on purely
classical data (e.g., a description of the system's state before measurement),
an extended relation which remains valid in the presence of quantum information
has been proposed recently [Berta et al., Nat. Phys. 6, 659 (2010)]. Here, we
generalize this uncertainty relation to one formulated in terms of smooth
entropies. Since these entropies measure operational quantities such as
extractable secret key length, our uncertainty relation is of immediate
practical use. To illustrate this, we show that it directly implies security of
a family of quantum key distribution protocols including BB84. Our proof
remains valid even if the measurement devices used in the experiment deviate
arbitrarily from the theoretical model.
