  We review recent progress on constructing non-equilibrium steady state
density operators of boundary driven locally interacting quantum chains, where
driving is implemented via Markovian dissipation channels attached to the
chain's ends. We discuss explicit solutions in three different classes of
quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg
spin-1/2 chain, the Fermi-Hubbard chain, and the Lai-Sutherland spin-1 chain,
and discuss universal concepts which characterize these solutions, such as
matrix product ansatz and a more structured walking graph state ansatz. The
central theme is the connection between the matrix product form of
nonequilibrium states and the integrability structures of the bulk Hamiltonian,
such as the Lax operators and the Yang-Baxter equation.
  However, there is a remarkable distinction with respect to the conventional
quantum inverse scattering method, namely addressing nonequilibrium steady
state density operators requires non-unitary irreducible representations of
Yang-Baxter algebra which are typically of infinite dimensionality. Such
constructions result in non-Hermitian, and often also non-diagonalisable
families of commuting transfer operators which in turn result in novel
conservation laws of the integrable bulk Hamiltonians. For example, in the case
of anisotropic Heisenberg model, quasi-local conserved operators which are odd
under spin reversal (or spin flip) can be constructed, whereas the conserved
operators stemming from orthodox Hermitian transfer operators (via logarithmic
differentiation) are all even under spin reversal.
