Stationary states of boundary driven quantum systems: some exact results

We study finite-dimensional open quantum systems whose density matrix evolves via a Lindbladian, ρ̇=-i[H,ρ]+ Dρ. Here H is the Hamiltonian of the isolated system and D is the dissipator. We consider the case where the system consists of two parts, the "boundary'' A and the ``bulk'' B, and D acts only on A, so D= DA⊗ IB, where DA acts only on part A, while IB is the identity superoperator on part B. Let DA be ergodic, so DAρ̂A=0 only for one unique density matrix ρ̂A. We show that any stationary density matrix ρ̄ on the full system which commutes with H must be of the product form ρ̄=ρ̂A⊗ρB for some ρB. This rules out finding any DA that has the Gibbs measure ρ_β~ e-β H as a stationary state with β≠ 0, unless there is no interaction between parts A and B. We give criteria for the uniqueness of the stationary state ρ̄ for systems with interactions between A and B. Related results for non-ergodic cases are also discussed.