  We study the time evolution of single-particle quantum states described by a
Lindblad master equation with local terms. By means of a geometric resolvent
equation derived for Lindblad generators, we establish a finite-volume-type
criterion for the decay of the off-diagonal matrix elements in the position
basis of the time-evolved or steady states. This criterion is shown to yield
exponential decay for systems where the non-hermitian evolution is either
gapped or strongly disordered. The gap exists for example whenever any level of
local dephasing is present in the system. The result in the disordered case can
be viewed as an extension of Anderson localization to open quantum systems.
