University College London
Scrambling of quantum information in unitary evolution can be hindered due to measurements and localization. Both these effects lead to pinning of the quantum mechanical wavefunction resulting in suppression of entanglement in the steady state. In monitored free-fermionic models the steady state undergoes an entanglement transition from a critical logarithmically to area-law entangled steady state due to the coupling to an environment. However, in an isolated system arbitrarily weak disorder in one dimension leads to Anderson localization. We investigate a free-fermion system in a random field subject to continuous monitoring, which enables us to probe the non-trivial interplay between measurement-induced phases and disorder. Through the careful analysis of the effective central charge, entanglement entropy, and density-density correlations, we show that the critical phase with conformal symmetry is stable under disorder perturbations until a finite critical disorder strength. We find that the universality class of the transition at finite disorder and dissipative coupling is consistent with the Berezinskii-Kosterlitz-Thouless across the extended phase diagram. Furthermore, destructive interference responsible for Anderson localization is destroyed under finite monitoring strength and the steady state orbital wavefunction exhibits a power-law decay. Our results indicate that the critical phase is robust to disorder and the area-law phase is distinct from Anderson localization at weak dissipation. Our work opens the avenue to probe this interesting phase transition in experiments involving electrons in quantum dot arrays and nanowires, and allow quantum control of entangled states of electrons.