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Explicit matrix product ansatz is presented, in first two orders in the(weak) coupling parameter, for the non-equilibrium steady state of thehomogeneous, nearest neighbor Heisenberg XXZ spin-1/2 chain driven by Lindbladoperators which act only at the edges of the chain. The first order of thedensity operator becomes in thermodynamic limit an exact pseudo-localconservation law and yields -- via Mazur inequality -- a rigorous lower boundon the high temperature spin Drude weight. Such Mazur bound is a non-vanishingfractal function of the anisotropy parameter Delta for |Delta|<1.

Open XXZ spin chain: Nonequilibrium steady state and strict bound on ballistic transport