  We present a general construction of matrix product states for stationary
density matrices of one-dimensional quantum spin systems kept out of
equilibrium through boundary Lindblad dynamics. As an application we review the
isotropic Heisenberg quantum spin chain which is closely related to the
generator of the simple symmetric exclusion process. Exact and heuristic
results as well as numerical evidence suggest a local quantum equilibrium and
long-range correlations reminiscent of similar large-scale properties in
classical stochastic interacting particle systems that can be understood in
terms of fluctuating hydrodynamics.
