  How violently do two quantum operators disagree? Different fields of physics
feature different measures of incompatibility: (i) In quantum information
theory, entropic uncertainty relations constrain measurement outcomes. (ii) In
condensed matter and high-energy physics, the out-of-time-ordered correlator
(OTOC) signals scrambling, the spread of information through many-body
entanglement. We unite these measures, proving entropic uncertainty relations
for scrambling. The entropies are of distributions over weak and strong
measurements' possible outcomes. The weak measurements ensure that the OTOC
quasiprobability (a nonclassical generalization of a probability, which
coarse-grains to the OTOC) governs terms in the uncertainty bound. The
quasiprobability causes scrambling to strengthen the bound in numerical
simulations of a spin chain. This strengthening shows that entropic uncertainty
relations can reflect the type of operator disagreement behind scrambling.
Generalizing beyond scrambling, we prove entropic uncertainty relations
satisfied by commonly performed weak-measurement experiments. We unveil a
physical significance of weak values (conditioned expectation values): as
governing terms in entropic uncertainty bounds.
