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  • Open Access

Diagnosing Chaos with Projected Ensembles of Process Tensors

Peter O’Donovan1,*, Neil Dowling2,3,†, Kavan Modi4,3,‡, and Mark T. Mitchison1,5,§

  • *Contact author: peodonov@tcd.ie
  • †Contact author: ndowling@uni-koeln.de
  • ‡Contact author: kavan@quantumlah.org
  • §Contact author: mark.mitchison@kcl.ac.uk
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PRX Quantum 7, 020322 – Published 5 May, 2026

DOI: https://doi.org/10.1103/fgc4-hgk1

Abstract

The process tensor provides a general representation of a quantum system evolving under repeated interventions and is fundamental for numerical simulations of local many-body dynamics. In this work, we introduce the projected process ensemble, an ensemble of pure output states of a process tensor in a given basis of local interventions, and use it to define increasingly more fine-grained probes of quantum chaos. The first moment of this ensemble encapsulates numerous previously studied chaos quantifiers, including the Alicki-Fannes quantum dynamical entropy, butterfly flutter fidelity, and spatiotemporal entanglement. We discover characteristic entanglement structures within the ensemble’s higher moments that can sharply distinguish chaotic from integrable dynamics, overcoming deficiencies of the quantum dynamical and spatiotemporal entropies. These conclusions are supported by extensive numerical simulations of many-body dynamics for a range of spin-chain models, including noninteracting, interacting-integrable, chaotic, and many-body localized regimes. Our work elucidates the fingerprints of chaos on spatiotemporal correlations in quantum stochastic processes and provides a unified framework for analyzing the complexity of unitary and monitored many-body dynamics.

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Popular Summary

In quantum systems, chaos is often defined by how random they appear, but no universal measure of this randomness exists, leading to many distinct approaches to quantifying chaos. We introduce the projected process ensemble (PPE) as a tool that quantifies chaos by measuring the complexity of the states produced after performing a sequence of local interventions on a many-body system. The PPE unifies several existing chaos indicators and reveals how chaotic systems generate near-maximal complexity, while nonchaotic systems do not. This framework provides a powerful way to study quantum chaos and analyze the temporal complexity of many-body dynamics.

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PRX Quantum 7, 020322– Published 5 May, 2026
  • Received 10 April 2025
  • Accepted 6 January 2026
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