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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.06887 |
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Table of Contents:
- We study finite-dimensional open quantum systems whose density matrix evolves via a Lindbladian, $\dotρ=-i[H,ρ]+{\mathcal D}ρ$. 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 ${\mathcal D}_A\hatρ_A=0$ only for one unique density matrix $\hatρ_A$. We show that any stationary density matrix $\barρ$ on the full system which commutes with $H$ must be of the product form $\barρ=\hatρ_A\otimesρ_B$ for some $ρ_B$. This rules out finding any ${\mathcal D}_A$ that has the Gibbs measure $ρ_β\sim e^{-βH}$ as a stationary state with $β\neq 0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\barρ$ for systems with interactions between $A$ and $B$. Related results for non-ergodic cases are also discussed.