  I employ random-matrix methods to set up and solve statistical models of
noisy nonunitary dynamics that appear in the context of monitored quantum
systems. The models cover a range of scenarios combining random dynamics and
measurements of variable strength of one or several qubits. The combined
dynamics drive the system into states whose statistics reflect the competition
of randomizing unitary evolution and the measurement-induced backaction
collapsing the state. These effects are mediated by entanglement, as I describe
in detail by analytical results. For the paradigmatic case of monitoring via a
single designated qubit, this reveals a simple statistical mechanism, in which
the monitoring conditions the state of the monitored qubit, which then imposes
statistical constraints on the remaining quantities of the system. For the case
of monitoring several qubits with prescribed strength, the developed formalism
allows one to set up the statistical description and solve it numerically.
Finally, I also compare the analytical results to the monitored dynamics of a
quantum kicked top, revealing two regimes where the statistical model either
describes the full stationary dynamics, or resolves time scales during
particular parts of the evolution.
