Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer

TL;DR

This paper models a measurement-induced phase transition in a chaotic dynamical system, revealing a transition between chaotic and bounded uncertainty phases, with exact solutions and numerical validation.

Contribution

It introduces a solvable model linking measurement effects in chaotic systems to directed polymer theory, providing exact critical behavior and a universal scaling function.

Findings

Identifies a phase transition in classical chaos due to measurements

Maps the problem to directed polymer on Cayley tree

Provides exact scaling function for entropy near criticality

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 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.