  We demonstrate that the exact non-equilibrium steady state of the
one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators
can be constructed explicitly with a matrix product ansatz for the
non-equilibrium density matrix where the matrices satisfy a {\it quadratic
algebra}. This algebra turns out to be related to the quantum algebra
$U_q[SU(2)]$. Coherent state techniques are introduced for the exact solution
of the isotropic Heisenberg chain with and without quantum boundary fields and
Lindblad terms that correspond to two different completely polarized boundary
states. We show that this boundary twist leads to non-vanishing stationary
currents of all spin components. Our results suggest that the matrix product
ansatz can be extended to more general quantum systems kept far from
equilibrium by Lindblad boundary terms.
