Measuring the distance between quantum many-body wave functions

Abstract

We study the distance of two wave functions under chaotic time evolution. The two initial states are differed only by a local perturbation. To be entitled "chaos" the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 (t) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 (t) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 (t) to the square of commutator C(t) (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A. Among various one-dimensional spin-1 2 models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 (t) when the local perturbation and subsystem A are well separated. This regime is absent in the spin-1 2 chain with local interaction only. After sufficiently long time evolution, d 2 (t) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 (t) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions. arXiv:1712.06054v2 [cond-mat.stat-mech]

Signature of quantum chaos in operator entanglement in 2d CFTs

“…One of our primary goals in this paper is to see if we are able to characterize chaotic behaviors of these CFTs by using operator entanglement measures. This is complimentary to and should be contrasted with prior works which tried to detect the signature of quantum chaos in CFTs by computing OTOCs [33,34,35,36], the relative entropy [37], and others [38,31]. (We compare these different indicators of scrambling and chaos in Sec.…”

“…• The n-th tri-partite operator mutual information (TOMI) These operator entanglement measures allow us to focus on the properties of the evolution operator itself and study their information scrambling and chaotic behaviors, independent of the choice of initial states of the time-evolution. For previous studies on the operator entanglement entropy of unitary evolution operators, see, for example, [28,29,30,31,32]. In particular, Ref.…”

Probing Scrambling Using Statistical Correlations between Randomized Measurements

“…At later times (middle panel), when ½V; WðtÞ ≠ 0 due to the spreading of WðtÞ, the value of hV † WðtÞVi u;k 0 becomes decorrelated from hWðtÞi u;k 0 . Note that in analogy to our approach, the distance between a quantum state and a (physical) copy, which is perturbed at t ¼ 0 by the operator V, has been proposed to numerically detect scrambling [54,55].…”

Quantum chaos dynamics in long-range power law interaction systems

“…Meanwhile, numerical calculation based on either exact diagonalization or matrix product approach 25 usually limits to small system size due to the large entanglement generated by chaotic dynamics. Additionally, long range interaction generates stronger finite size effect 26,27 compared to the models with short range interaction. To circumvent these difficulties, we construct a Brownian quantum circuit model with power law interaction.…”