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  • Open Access

Free Probability in a Minimal Quantum Circuit Model

Felix Fritzsch1,* and Pieter W. Claeys1,2

  • *Contact author: fritzsch@pks.mpg.de
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Phys. Rev. X 16, 031027 – Published 5 August, 2026

DOI: https://doi.org/10.1103/6mpz-p85s

Abstract

Recent experimental and theoretical developments in many-body quantum systems motivate the study of their out-of-equilibrium properties through multitime 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 timescales, 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. 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.

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Popular Summary

Understanding how quantum information spreads and gets scrambled is a central challenge in modern quantum science. These phenomena can be characterized through the dynamics of experimentally accessible (higher-order) out-of-time-order correlation functions (OTOCs). While scrambling has been linked to the mathematical framework of free probability, this connection has largely relied on numerical evidence and lacked a clear dynamical explanation.

In this work, we introduce a minimal model of many-body quantum dynamics in which the dynamics of higher-order OTOCs can be characterized exactly. We show that both the stationary state and the relaxation toward it are naturally described by free cumulants, key objects in free probability theory. The relaxation is understood as the emergence of free independence between time-evolved and static observables, revealing the organizing principle behind otherwise complex scrambling dynamics. Although derived in a minimal model, our results provide a framework for understanding information spreading in more realistic quantum many-body systems.

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Phys. Rev. X 16, 031027– Published 5 August, 2026
  • Received 24 June 2025
  • Revised 24 April 2026
  • Accepted 19 May 2026
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