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

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.