We study the 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 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 the dynamical properties of localization, which cannot be understood from the stationary properties of the entanglement entropy.

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• Dynamical phase transitions
• Many-body localization
• Nonequilibrium statistical mechanics
• Open quantum systems
• Quantum entanglement
• Quantum measurements
• Quantum stochastic processes
• Disordered systems
• Ultracold gases