We study finite-dimensional open quantum systems whose density matrix evolves via a Lindbladian, $\dot{ \rho}=-i[H,\rho]+{ \mathcal D}\rho$. Here $H$ is the Hamiltonian of the isolated system and ${ \mathcal D}$ is the dissipator. We consider the case where the system consists of two parts, the "boundary'' $A$ and the ``bulk'' $B$, and ${ \mathcal D}$ acts only on $A$, so ${ \mathcal D}={ \mathcal D}_A\otimes{ \mathcal I}_B$, where ${ \mathcal D}_A$ acts only on part $A$, while ${ \mathcal I}_B$ is the identity superoperator on part $B$. Let ${ \mathcal D}_A$ be ergodic, so ${ \mathcal D}_A\hat{ \rho}_A=0$ only for one unique density matrix $\hat{ \rho}_A$. We show that any stationary density matrix $\bar{ \rho}$ on the full system which commutes with $H$ must be of the product form $\bar{ \rho}=\hat{ \rho}_A\otimes\rho_B$ for some $\rho_B$. This rules out finding any ${ \mathcal D}_A$ that has the Gibbs measure $\rho_\beta\sim e^{ -\beta H}$ as a stationary state with $\beta\neq 0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\bar{ \rho}$ for systems with interactions between $A$ and $B$. Related results for non-ergodic cases are also discussed.
Categories: quant-ph