  The entropic uncertainty principle in the form proven by Maassen and Uffink
yields a fundamental inequality that is prominently used in many places all
over the field of quantum information theory. In this work, we provide a family
of versatile generalizations of this relation. Our proof methods build on a
deep connection between entropic uncertainties and interpolation inequalities
for the doubly stochastic map that links probability distributions in two
measurements bases. In contrast to the original relation, our generalization
also incorporates the von Neumann entropy of the underlying quantum state.
These results can be directly used to bound the extractable randomness of a
source independent QRNG in the presence of fully quantum attacks, to certify
entanglement between trusted parties, or to bound the entanglement of a system
with an untrusted environment.
