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In this short note we provide two extensions on the recent explicit resultson the matrix-product ansatz for the non-equilibrium steady state of amarkovianly boundary-driven anisotropic Heisenberg XXZ spin 1/2 chain. We writea perturbative solution for the steady state density matrix in the system-batyhcoupling for an arbitrary (asymmetric) set of four spin-flip rates at the twochain ends, generalizing the symmetric-driving ansatz of [Phys. Rev. Lett. 106,217206 (2011)]. Furthermore, we generalize the exact (non-perturbative) form ofthe steady state for just two Lindblad channels (spin-up flipping on the left,and spin-down flipping on the right) to an arbitrary (asymmetric) ratio of thespin flipping rates [Phys. Rev. Lett. 107, 137201 (2011)]. In addition, we alsoindicate a simple proof of uniqueness of our steady states.

Comments on boundary driven open XXZ chain: asymmetric driving and uniqueness of steady states