Classical stochastic dynamics and quantum unitary dynamics are limits of the same system. Between them sits monitored quantum dynamics — quantum evolution interrupted by measurements. The measurement strength is the tuning parameter: at zero measurement, the system evolves unitarily (fully quantum). At strong measurement, it collapses to classical stochastic dynamics (the quantum Zeno regime, where constant measurement forces classical trajectories).
The East circuit model makes this interpolation explicit. Based on kinetically constrained systems — lattice models where a site can flip only if its neighbor is in the right state — the classical version has well-understood dynamical phase coexistence: an active phase (many flips, high activity) and an inactive phase (frozen, low activity) coexist in the same system at different space-time regions. The quantum version inherits this structure but modifies it as measurement strength varies.
The key idea is treating space-time measurement records as states in a fictitious spin system and applying thermodynamic methods to characterize the dynamical phases. This converts a question about quantum dynamics into a question about thermodynamics of an auxiliary classical system. The phases of the quantum dynamics become phases of the fictitious spin model.
The structural through-claim: measurement does not simply destroy quantum dynamics and replace it with classical dynamics. Instead, it continuously deforms one into the other, preserving the structure of dynamical phase coexistence throughout the interpolation. The phases survive the transition from quantum to classical — they are not features of one regime that vanish in the other but features of the dynamics that transform smoothly as the measurement rate changes. The measurement is a bridge, not a wall.
(arXiv:2603.18227)