  Quantum systems with a non-conserved probability can be described by means of
non-Hermitian Hamiltonians and non-unitary dynamics. In this paper, the case in
which the degrees of freedom can be partitioned in two subsets with light and
heavy masses is treated. A classical limit over the heavy coordinates is taken
in order to embed the non-unitary dynamics of the subsystem in a classical
environment. Such a classical environment, in turn, acts as an additional
source of dissipation (or noise), beyond that represented by the non-unitary
evolution. The non-Hermitian dynamics of a Heisenberg two-spin chain, with the
spins independently coupled to harmonic oscillators, is considered in order to
illustrate the formalism.
