We study composite open quantum systems with a finite-dimensional state space \(\mathcal {H}_{AB} = \mathcal {H}_A\otimes \mathcal {H}_B\) governed by a Lindblad equation \(\displaystyle {\frac{\textrm{d}}{\textrm{d}t}\rho (t) = \mathcal {L}_\gamma \rho (t)}\) where \(\mathcal {L}_\gamma \rho = -i[H,\rho ] + \gamma \mathcal {D}\rho \) . Here, H is a Hamiltonian on \(\mathcal {H}_{AB}\) while \(\mathcal {D}\) is a dissipator \(\mathcal {D}_A\otimes \mathbbm {1}\) acting non-trivially only on part A of the system, which can be thought of as the boundary, and \(\gamma \) is a parameter. It is known that the dynamics may simplify as the Zeno limit, \(\gamma \rightarrow \infty \) , is approached, so that after a initial time of order \(\gamma ^{-1}\) , \(\rho (t)\) is well approximated by \(\pi _A\otimes R(t)\) where \(\pi _A\) is a density matrix on \(\mathcal {H}_A\) such that \(\mathcal {D}_A\pi _A =0\) , and R(t) is an approximate solution of \({\displaystyle \frac{\textrm{d}}{\textrm{d}t}R(t) = \mathcal {L}_{P,\gamma }R(t)}\) where \(\mathcal {L}_{P,\gamma } R:= -i[H_P,R] + \gamma ^{-1} \mathcal {D}_P R\) with \(H_P\) being a Hamiltonian on \(\mathcal {H}_B\) and \(\mathcal {D}_P\) being a Lindblad generator acting on density matrices on \(\mathcal {H}_B\) . We give a rigorous proof of this holding in greater generality than in previous work; we assume only that \(\mathcal {D}_A\) is ergodic and gapped. Moreover, we precisely control the error terms, and use this to show that the mixing times of \(\mathcal {L}_\gamma \) and \(\mathcal {L}_{P,\gamma }\) are tightly related near the Zeno limit. Despite this connection, the errors in the approximate description of the evolution accumulate on times of order \(\gamma ^2\) , so it is difficult to directly access steady states \(\bar{\rho }_\gamma \) of \(\mathcal {L}_\gamma \) through study of \(\mathcal {L}_{P,\gamma }\) . In order to better control the long time behavior, and in particular the steady states \(\bar{\rho }_\gamma \) , we introduce a third Lindblad generator \(\mathcal {D}_P^\sharp \) that does not involve \(\gamma \) , but is still closely related to \(\mathcal {L}_\gamma \) and \(\mathcal {L}_{P,\gamma }\) . We show that if \(\mathcal {D}_P^\sharp \) is ergodic and gapped, then so are \(\mathcal {L}_\gamma \) and \(\mathcal {L}_{P,\gamma }\) for all large \(\gamma \) , and in this case, if \(\bar{\rho }_\gamma \) denotes the unique steady state for \(\mathcal {L}_\gamma \) , then \(\lim _{\gamma \rightarrow \infty }\bar{\rho }_\gamma = \pi _A\otimes \bar{R}\) where \(\bar{R}\) is the unique steady state for \(\mathcal {D}_P^\sharp \) . We further show that there is a trace norm convergent expansion \({\displaystyle \bar{\rho }_\gamma = \pi _A\otimes \bar{R} +\gamma ^{-1} \sum _{k=0}^\infty \gamma ^{-k} \bar{n}_k}\) where, defining \(\bar{n}_{-1}:= \pi _A\otimes \bar{R}\) , \(\mathcal {D}\bar{n}_k = -i[H,\bar{n}_{k-1}]\) for all \(k\geqslant 0\) . Using properties of \(\mathcal {D}_P\) and \(\mathcal {D}_P^\sharp \) , we show that this system of equations has a unique solution. This is illustrated in a simple example for which one can compute \(\bar{\rho }_\gamma \) , and can carry out the expansion explicitly.
