Emergent deterministic entanglement dynamics in monitored infinite-range bosonic systems

We study monitored quantum dynamics of infinite-range interacting bosonic systems in the thermodynamic limit. We show that under semiclassical assumptions, the quantum fluctuations along single monitored trajectories adopt a deterministic limit for both quantum-jump and state-diffusion unravelings, and can be exactly solved. In particular, the hierarchical structure of the equations of motion explains the coincidence of entanglement phase transitions and dissipative phase transitions found in previous finite-size numerical studies. We illustrate the findings on a Bose-Hubbard dimer and a collective spin system.