  Scrambling of quantum information in unitary evolution can be hindered due to
measurements and localization, which pin quantum mechanical wavefunctions in
real space suppressing entanglement in the steady state. In monitored
free-fermionic models, the steady state undergoes an entanglement transition
from a logarithmically entangled critical state to area-law. However, disorder
can lead to Anderson localization. We investigate free fermions in a random
potential with continuous monitoring, which enables us to probe the interplay
between measurement-induced and localized phases. We show that the critical
phase is stable up to a finite disorder and the criticality is consistent with
the Berezinskii-Kosterlitz-Thouless universality. Furthermore, monitoring
destroys localization, and the area-law phase at weak dissipation exhibits
power-law decay of single-particle wave functions. Our work opens the avenue to
probe this novel phase transition in electronic systems of quantum dot arrays
and nanowires, and allow quantum control of entangled states.
