Interface Pinning and an Entanglement Transition in Weakly-Monitored Quantum Dynamics

Weakly-monitored quantum dynamics -- involving local unitary evolution and infrequent projective measurements -- are known to produce volume-law entangled steady-states that are good at "hiding" local quantum information. Here we show that transitions between this phase, and a phase in which the steady-state resembles a more conventional, volume-law entangled (Page) state can arise in local, weakly-monitored dynamics. We argue that in one spatial dimension, this transition is related to a "pinning" phase transition for a directed polymer in a random environment and in the presence of an attractive interface. We compare our predictions for the critical scaling of the entanglement and for certain properties of the two phases in this setting to results obtained from large-scale numerical simulations of the transition in Clifford circuit dynamics. We also discuss predictions for this phase transition in higher dimensions, and how the classical descriptions of these entanglement transitions can also be used to understand the effects of "unrecorded" measurements on the entanglement growth in quantum many-body dynamics.