  Markovian reservoir engineering, in which time evolution of a quantum system
is governed by a Lindblad master equation, is a powerful technique in studies
of quantum phases of matter and quantum information. It can be used to drive a
quantum system to a desired (unique) steady state, which can be an exotic phase
of matter difficult to stabilize in nature. It can also be used to drive a
system to a unitarily-evolving subspace, which can be used to store, protect,
and process quantum information. In this paper, we derive a formula for the map
corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to
study several important features of the unique state and subspace cases. We
quantify how subspaces retain information about initial states and show how to
use Lindbladians to simulate any quantum channels. We show that the quantum
information in all subspaces can be successfully manipulated by small
Hamiltonian perturbations, jump operator perturbations, or adiabatic
deformations. We provide a Lindblad-induced notion of distance between
adiabatically connected subspaces. We derive a Kubo formula governing linear
response of subspaces to time-dependent Hamiltonian perturbations and determine
cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an
application, we show that (for gapped systems) the zero-frequency Hall
conductivity is unaffected by many types of Markovian dissipation. Finally, we
show that the energy scale governing leakage out of the subspaces, resulting
from either Hamiltonian/jump-operator perturbations or corrections to adiabatic
evolution, is different from the conventional Lindbladian dissipative gap and,
in certain cases, is equivalent to the excitation gap of a related Hamiltonian.
