In a closed single-particle quantum system, spatial disorder induces Anderson
localization of eigenstates and halts wave propagation. The phenomenon is
vulnerable to interaction with environment and decoherence, that is believed to
restore normal diffusion. We demonstrate that for a class of experimentally
feasible non-Hermitian dissipators, which admit signatures of localization in
asymptotic states, quantum particle opts between diffusive and ballistic
regimes, depending on the phase parameter of dissipators, with sticking about
localization centers. In diffusive regime, statistics of quantum jumps is
non-Poissonian and has a power-law interval, a footprint of intermittent
locking in Anderson modes. Ballistic propagation reflects dispersion of an
ordered lattice and introduces a new timescale for jumps with non-monotonous
probability distribution. Hermitian dephasing dissipation makes localization
features vanish, and Poissonian jump statistics along with normal diffusion are
recovered.Comment: 6 pages, 5 figure
