We construct an explicit matrix product ansatz for the steady state of a boundary driven XYZ spin-$\frac{1}{2}$ chain for arbitrary local polarizing channels at the ends of the chain. The ansatz, where the Lax operators are written explicitly in terms of infinite-dimensional bidiagonal (triangular) site-dependent matrices, becomes exact either in the (Zeno) limit of infinite dissipation strength or the thermodynamic limit of infinite chain length. The solution is based on an extension of the recently discovered family of separable eigenstates of the model.

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• Nonequilibrium statistical mechanics
• Exact solutions for many-body systems
• Matrix product states
• Quantum spin chains