  We study the dynamics of a monitored single particle in a one-dimensional,
Anderson-localized system. The time evolution is governed by Hamiltonian
dynamics for fixed time intervals, interrupted by local, projective
measurements. The competition between disorder-induced localization and
measurement-induced jumps leads to interesting behaviour of readout-averaged
quantities. We find that measurements at random positions delocalize the
average position, similar to a classical random walk. Along each quantum
trajectory, the particle remains localized, however with a modified
localization length. In contrast to measurement-induced delocalization,
controlled measurements can be used to introduce transport in the system and
localize the particle at a chosen site. In this sense, the measurements provide
a controlled environment for the particle.
