  Monitored quantum many-body systems display a rich pattern of entanglement
dynamics, which is unique to this non-unitary 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. We consider two
examples, a monitored SSH model and a quantum Ising chain, for which we show
the jumps have remarkably different effects on the entanglement despite having
the same statistics as encoded in their waiting-time distribution. To
understand this difference, we introduce a new metric, the statistics of
entanglement gain and loss due to jumps and non-Hermitian evolution. 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 demonstrate that significant deviations from the no-click limit
arise whenever quantum jumps strongly renormalize the non-Hermitian dynamics,
as in the case of the SSH model at weak monitoring or in the Ising chain at
large transverse field. On the other hand, we show that the weak monitoring
phase of the Ising chain leads to a robust sub-volume logarithmic phase due to
weakly renormalized non-Hermitian dynamics.
