  We study localization properties of continuously monitored dynamics and
associated measurement-induced phase transitions in disordered quantum
many-body systems on the basis of the quantum trajectory approach. By
calculating the fidelity between random quantum trajectories, we demonstrate
that the disorder and the measurement can lead to dynamical properties distinct
from each other, although both have a power to suppress the entanglement
spreading. In particular, in the large-disorder regime with weak measurement,
we elucidate that the fidelity exhibits an anomalous power-law decay before
saturating to the steady-state value. Furthermore, we propose a general method
to access physical quantities for quantum trajectories in continuously
monitored dynamics without resorting to postselection. It is demonstrated that
this scheme drastically reduces the cost of experiments. Our results can be
tested in ultracold atoms subject to continuous measurement and open the avenue
to study dynamical properties of localization, which cannot be understood from
the stationary properties of the entanglement entropy.
