Free Probability in a Minimal Quantum Circuit Model

TL;DR

This paper analyzes the decay and dynamics of higher-order out-of-time-order correlators in a minimal quantum circuit model, linking free probability and quantum memory effects.

Contribution

It introduces a model capturing environment effects on local quantum systems using influence matrices and connects OTOC dynamics with free probability theory.

Findings

All higher-order OTOCs decay exponentially over time.

Local operators tend toward free independence at late times.

The influence matrix provides a Markovian description of the dynamics.

Abstract

Recent experimental and theoretical developments in many-body quantum systems motivate the study of their out-of-equilibrium properties through multi-time correlation functions. We consider the dynamics of higher-order out-of-time-order correlators (OTOCs) in a minimal circuit model for quantum dynamics. This model mimics the dynamics of a structured subsystem locally coupled to a maximally random environment. We prove the exponential decay of all higher-order OTOCs and fully characterize the relevant time scales, showing how local operators approach free independence at late times. We show that the effects of the environment on the local subsystem can be captured in a higher-order influence matrix, which allows for a Markovian description of the dynamics provided an auxiliary degree of freedom is introduced. This degree of freedom directly yields a dynamical picture for the OTOCs in terms of free cumulants from free probability, consistent with recent predictions from the full eigenstate thermalization hypothesis (ETH). This approach and the relevant influence matrix are expected to be applicable in more general settings and present a first step to characterizing quantum memory in higher-order OTOCs.