Entropic uncertainty relations for quantum information scrambling

Abstract

Different fields of physics characterize differently how much two quantum operations disagree: quantum information theory features uncertainty relations cast in terms of entropies. The higher an uncertainty bound, the less compatible the operations. In condensed matter and high-energy physics, initially localized, far-apart operators come to disagree as entanglement spreads through a quantum many-body system. This spread, called "scrambling," is quantified with the out-of-time-ordered correlator (OTOC). We unite these two measures of operation disagreement by proving entropic uncertainty relations for scrambling. The uncertainty bound depends on the quasiprobability (the nonclassical generalization of a probability) known to average to the OTOC. The quasiprobability strengthens the uncertainty bound, we find, when a spin chain scrambles in numerical simulations. Hence our entropic uncertainty relations reflect the same incompatibility as scrambling, uniting two fields' notions of quantum-operation disagreement.

Weak Measurement of a Superconducting Qubit Reconciles Incompatible Operators

“…We demonstrate a new physical meaning of the weak value: As a contribution to the uncertainty bound for weak and strong measurements [22], A wv controls how much weak measurements reconcile incompatibility. This Letter reports on an experimental test of the entropic uncertainty relation for weak and strong measurements [22]. We first introduce the experimental platform and the dispersive measurements performed in circuit QED.…”

“…An entropic uncertainty relation that governs weak measurements was proved recently [22]. The relation quantifies the disagreement between a strong measurement and the composition of a weak measurement and another strong measurement.…”

“…We assume, throughout this Letter, that the eigenspaces are nondegenerate, as we will focus on a qubit. But this formalism, and the theory we test [22], extend to degeneracies. Consider measuring I, obtaining outcome λ i , measuring F, and obtaining outcome f. Let A denote an observable that commutes with neither I nor F. Which value can be retrodictively ascribed most reasonably to A, given the preselection on λ i and the postselection on f?…”

“…But, under a natural normalization scheme, the weak measurement reduces the sum of the two operations' entropies. The entropy sum was bounded in [22], whose theory we review and then test experimentally. We quantify how the weak measurement backacts on the state.…”

“…In bridging entropic uncertainty relations with weak measurements and superconducting qubits, this Letter unites several subfields of quantum-information physics, which can benefit from the synergy introduced here. For example, in addition to the fundamental contribution to weak values mentioned above, this Letter paves the path toward detecting quantum chaos with weak measurements experimentally, being the first experiment to sprout from the considerable theoretical work on leveraging weak measurements to identify chaos [22,[30][31][32][33][34][35][36][37].…”