Entropic uncertainty relations for quantum information scrambling

Abstract

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.