Entanglement Crossovers Under Time-Dependent U(1) Symmetry Breaking in Monitored Quantum Circuits

Conservation laws reshape quantum information dynamics by restricting how scrambling can occur. In monitored circuits, this raises a natural dynamical question: when a symmetry constraint is gradually relaxed, how quickly do the entanglement dynamics forget the symmetric regime? We address this question in one-dimensional monitored random circuits by driving the two-qubit gate ensemble from exact U(1) conservation toward unconstrained Haar dynamics. The response is not instantaneous: the entanglement retains a finite-depth memory of the conserving dynamics and crosses over toward the unconstrained behavior over a rate-dependent time window. Operational 25%, 50%, and 75% crossover times all move to later circuit depths for slower drives. At p=0.4, the same rate ordering is also observed through L=32 in a finite-time matrix-product-state consistency check with a limited trajectory ensemble. By contrast, the weak fixed perturbations studied here, ϵ(t)=ϵ0≤0.1, remain close to the conserving reference over the same finite observation window. Within this finite-time window, the driven crossover is therefore not reproduced merely by waiting under the fixed perturbations tested. Purification dynamics from a maximally mixed state exhibit the same qualitative response. We therefore interpret the results as a finite-time dynamical crossover from symmetry-constrained dynamics toward less constrained scrambling, rather than as evidence for a universal crossover exponent or a microscopic noise law. The study provides a controlled setting for asking how long symmetry-dependent information-processing behavior persists as a conservation law is deliberately weakened.