Quasiprobability behind the out-of-time-ordered correlator

Abstract

Two topics, evolving rapidly in separate fields, were combined recently: The out-of-time-ordered correlator (OTOC) signals quantum-information scrambling in many-body systems. The Kirkwood-Dirac (KD) quasiprobability represents operators in quantum optics. The OTOC was shown to equal a moment of a summed quasiprobability [Yunger Halpern, Phys. Rev. A 95, 012120 (2017)]. That quasiprobability, we argue, is an extension of the KD distribution. We explore the quasiprobability's structure from experimental, numerical, and theoretical perspectives. First, we simplify and analyze Yunger Halpern's weak-measurement and interference protocols for measuring the OTOC and its quasiprobability. We decrease, exponentially in system size, the number of trials required to infer the OTOC from weak measurements. We also construct a circuit for implementing the weak-measurement scheme. Next, we calculate the quasiprobability (after coarse-graining) numerically and analytically: We simulate a transverse-field Ising model first. Then, we calculate the quasiprobability averaged over random circuits, which model chaotic dynamics. The quasiprobability, we find, distinguishes chaotic from integrable regimes. We observe nonclassical behaviors:The quasiprobability typically has negative components. It becomes nonreal in some regimes. The onset of scrambling breaks a symmetry that bifurcates the quasiprobability, as in classical-chaos pitchforks. Finally, we present mathematical properties. We define an extended KD quasiprobability that generalizes the KD distribution. The quasiprobability obeys a Bayes-type theorem, for example, that exponentially decreases the memory required to calculate weak values, in certain cases. A time-ordered correlator analogous to the OTOC, insensitive to quantum-information scrambling, depends on a quasiprobability closer to a classical probability. This work not only illuminates the OTOC's underpinnings, but also generalizes quasiprobability theory and motivates immediate-future weak-measurement challenges.2. Bayes-type theorem and retrodiction with the KD quasiprobability chain example. Discrete systems are governed by quasiprobabilities, which resemble probabilities. Continuous systems are governed by quasiprobability densities, which resemble probability densities. Our quasiprobabilities can be replaced with quasiprobability densities, and our sums can be replaced with integrals, in, e.g., quantum field theory.

Resilience of scrambling measurements

“…However, averaging over weakmeasurement trials reproduces strong-measurement statistics. Also, weak measurements offer experimental access to OTOCs and to more-fundamental quasiprobabilities [11,12].…”

“…II of [12]. 1 We focus on the simplified protocol, though the renormalization scheme is expected to extend to the original protocol.…”

“…We focus on two scrambling protocols, the interferometric protocol [23] and the weak-measurement protocol [11,12], but we expect our results to extend to other OTOC measurement schemes. The paper is structured as follows.…”

Optimizing measurement strengths for qubit quasiprobabilities behind out-of-time-ordered correlators

“…The QPDp t formally consists of 2 4 complex numbers, so apparently it requires experimental determination of 32 real parameters. However, we can reduce this complexity to just 8 real parameters to measure [42]. Since A 2 = B 2 (t) = I, we use the identities Π A a = [I + (−1) a A]/2 and Π…”

“…Expanding upon the idea of the OTOC, we recently introduced a more refined and robust informationscrambling witness by decomposing the OTOC into its extended (coarse-grained) Kirkwood-Dirac [34][35][36][37][38][39][40] quasiprobability distribution (QPD) [41,42]. This QPD has since found utility in entropic uncertainty relations for scrambling [43], and is closely related to a witness for quantum advantage in postselected metrology [44].…”