Entanglement Dynamics in Monitored Systems and the Role of Quantum Jumps

Monitored quantum many-body systems display a rich pattern of entanglement dynamics, which is unique to this nonunitary setting. This work studies the effect of quantum jumps on the entanglement dynamics beyond the no-click limit corresponding to a deterministic non-Hermitian evolution. To this aim, we introduce a new tool that looks at the statistics of entanglement-entropy gain and loss after and in between quantum jumps. This insight allows us to build a simple stochastic model of a random walk with partial resetting, which reproduces the entanglement dynamics, and to dissect the mutual role of jumps and non-Hermitian evolution on the entanglement scaling. We apply these ideas to the study of measurement-induced transitions in monitored fermions. We demonstrate that significant deviations from the no-click limit arise whenever quantum jumps strongly renormalize the non-Hermitian dynamics, as in the case of models with $U(1)$ symmetry at weak monitoring. On the other hand, we show that the weak-monitoring phase of the Ising chain leads to a robust subvolume logarithmic phase due to weakly renormalized non-Hermitian dynamics.