  We consider an open quantum system, with dissipation applied only to a part
of its degrees of freedom, evolving via a quantum Markov dynamics. We
demonstrate that, in the Zeno regime of large dissipation, the relaxation of
the quantum system towards a pure quantum state is linked to the evolution of a
classical Markov process towards a single absorbing state. The rates of the
associated classical Markov process are determined by the original quantum
dynamics. Extension of this correspondence to absorbing states with internal
structure allows us to establish a general criterion for having a Zeno-limit
nonequilibrium stationary state of arbitrary finite rank. An application of
this criterion is illustrated in the case of an open XXZ spin-1/2 chain
dissipatively coupled at its edges to baths with fixed and different
polarizations. For this system, we find exact nonequilibrium steady-state
solutions of ranks 1 and 2.
