  We discuss many-body fermionic and bosonic systems subject to dissipative
particle losses in arbitrary spatial dimensions $d$, within the Keldysh
path-integral formulation of the quantum master equation. This open quantum
dynamics represents a generalisation of classical reaction-diffusion dynamics
to the quantum realm. We first show how initial conditions can be introduced in
the Keldysh path integral via boundary terms. We then study binary annihilation
reactions $A+A\to\emptyset$, for which we derive a Boltzmann-like kinetic
equation. The ensuing algebraic decay in time for the particle density depends
on the particle statistics. In order to model possible experimental
implementations with cold atoms, for fermions in $d=1$ we further discuss
inhomogeneous cases involving the presence of a trapping potential. In this
context, we quantify the irreversibility of the dynamics studying the time
evolution of the system entropy for different quenches of the trapping
potential. We find that the system entropy features algebraic decay for
confining quenches, while it saturates in deconfined scenarios.
