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Measurement-induced phase transition in state estimation of chaotic systems and the directed polymer

Federico Gerbino1,*, Guido Giachetti2, Pierre Le Doussal2, and Andrea De Luca3

  • *Contact author: federico.gerbino@universite-paris-saclay.fr
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Phys. Rev. Research 7, 033105 – Published 31 July, 2025

DOI: https://doi.org/10.1103/6375-8ncz

Abstract

We introduce a solvable model of a measurement-induced phase transition (MIPT) in a deterministic but chaotic dynamical system with a positive Lyapunov exponent. In this setup, an observer only has a probabilistic description of the system but mitigates chaos-induced uncertainty through repeated measurements. Using a minimal representation via a branching tree, we map this problem to the directed polymer (DP) model on the Cayley tree, although in a regime dominated by rare events. By studying the Shannon entropy of the probability distribution estimated by the observer, we demonstrate a phase transition distinguishing a chaotic phase with a reduced Lyapunov exponent from a strong-measurement phase where uncertainty remains bounded. Remarkably, the location of the MIPT transition coincides with the freezing transition of the DP, although the critical properties differ. We provide an exact, universal scaling function describing the entropy growth in the critical regime. Numerical simulations confirm our theoretical predictions, highlighting a simple yet powerful framework to explore measurement-induced transitions in classical chaotic systems.

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Phys. Rev. Research 7, 033105– Published 31 July, 2025
  • Received 15 January 2025
  • Accepted 21 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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