The Measurement-Driven Entanglement Phase Transition: Relation to Cluster Fragmentation

We study the dynamics of a collection of spins evolving under unitary gates and random projective measurements, and in the absence of spatial locality. For a certain choice of Clifford unitary gates, we map the wavefunction of the spins to the state of an evolving cluster, which undergoes a dynamical "fragmentation" transition as a function of the measurement rate. This classical dynamical transition corresponds to a phase transition in the asymptotic state of the spins; above a critical rate of applied measurements, the system exists in a product state over extensively many subsystems, while below this threshold, the wavefunction is no longer separable, and all subsystems develop volume-law entanglement. We show that the scaling of the entanglement entropy, which is related to the connectivity of the evolving cluster, distinguishes between the two phases, and identify a protocol to measure a local order parameter for this transition. The dynamics of the cluster may be studied analytically to determine other properties of the state of the spins, such as the distribution of stabilizer lengths. We discuss the relevance of our results to the measurement-driven entanglement transition that has been observed in higher dimensions.