We study composite open quantum systems with a finite-dimensional state space ${\cal H}_{AB} = {\cal H}_A\otimes {\cal H}_B$
governed by a Lindblad equation
$\rho'(t) = {\cal L}_\gamma \rho(t)$
where ${\cal L}_\gamma\rho = -i[H,\rho] + \gamma {\cal D} \rho$.
Here, $H$ is a Hamiltonian on ${\cal H}_{AB}$ while ${\cal D}$ is a dissipator ${\cal D}_A\otimes I$ 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 simplifies as the Zeno limit, $\gamma \to \infty$, is approached: 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 ${\cal H}_A$ such that ${\cal D}_A\pi_A =0$, and $R(t)$ is an approximate solution of
${\displaystyle R'(t) = {\cal L}_{P,\gamma}R(t)}$
where ${\cal L}_{P,\gamma} R := -i[H_P,R] + \gamma^{-1} {\cal D}_P R$
with $H_P$ being a Hamiltonian on ${\cal H}_B$ and ${\cal D}_P$ being a Lindblad generator acting on density matrices on ${\cal H}_B$. We give a rigorous proof of this holding in greater generality than in previous work; we assume only that ${\cal D}_A$ is ergodic and gapped.
Moreover, we precisely control the error terms, and use this to show
that the mixing times of ${\cal L}_\gamma$ and ${\cal L}_{P,\gamma}$ 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 ${\cal L}_\gamma$
through study of ${\cal L}_{P,\gamma}$. In order to better control the long
time behavior, and in particular the steady states $\bar\rho_\gamma$, we intoduce a third Lindblad generator
${\cal D}_P^\sharp$ that does not involve $\gamma$, but is still closely
related to ${\cal L}_\gamma$ and ${\cal L}_{P,\gamma}$. We show that if
${\cal D}_P^\sharp$ is ergodic and gapped, then so are ${\cal L}_\gamma$ and
${\cal L}_{P,\gamma}$
for all large $\gamma$. In this case, if $\bar\rho_\gamma$ denotes the
unique steady state for ${\cal L}_\gamma$, then
$\lim_{\gamma\to\infty}\bar\rho_\gamma = \pi_A\otimes \bar R$ where $\bar R$ is the unique steady state for
${\cal D}_P^\sharp$. 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 $\bar n_{-1} := \pi_A\otimes\bar R$, ${\cal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$.
Using properties of ${\cal D}_P$ and ${\cal D}_P^\sharp$, we show that this system of equations has a unique solution,
and prove convergence. This is illustrated in a simple example for which one can solve for $\bar\rho_\gamma$, and can carry out the expansion explicitly.
We apply these results to state preparation. This is joint work with David Huse and Joel Lebowitz.
Copyright. All Rights Reserved.