Stationary states of boundary-driven quantum systems: Some exact results

We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix ρ evolves via a Lindbladian, ρ˙=-i[H,ρ]+Dρ. Here H is the Hamiltonian of the 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 IB is the identity superoperator on part B. Let DA be ergodic, so DAπA=0 only for one unique density matrix πA. We show that any stationary density matrix ρ¯ on the full system which commutes with H must be of the product form ρ¯=πA - ρB for some ρB. This rules out finding any DA that has the Gibbs measure ρβ=e-βH/Z(β) 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 ρ¯ for systems with interactions between A and B. Related results for nonergodic cases are also discussed.