  We consider an extension of classical stochastic reaction-diffusion (RD)
dynamics to open quantum systems. We study a class of models of hard core
particles on a one-dimensional lattice whose dynamics is generated by a quantum
master operator and where particle hopping is coherent while reactions, such as
pair annihilation or pair coalescence, are dissipative. These are quantum open
generalisations of the $A+A \to \varnothing$ and $A+A \to A$ classical RD
models. We characterise the relaxation of the state towards the stationary
regime via a decomposition of the system Hilbert space into transient and
recurrent subspaces. We provide a complete classification of the structure of
the recurrent subspace (and the non-equilibrium steady states) in terms of the
dark states associated to the quantum master operator and its general spectral
properties. We also show that, in one dimension, relaxation towards these
absorbing dark states is slower than that predicted by a mean-field analysis
due to fluctuation effects, in analogy with what occurs in classical RD
systems. Numerical simulations of small systems suggest that the decay of the
density in one dimension, in both the open quantum $A+A \to \varnothing$ and
$A+A \to A$ cases, may go asymptotically as $t^{-b}$ with $1/2 < b < 1$.
