Proposition: A fixed autonomous finite proxy is recurrent

For a fixed finite-dimensional Hamiltonian $H$, fixed preparation $V_0$, and fixed access partition, the complete channel of $e^{-itH}{V}_0$ returns arbitrarily close to its initial channel at arbitrarily late times. If the initial accessible channel is replacement, then

Proof. There are finitely many eigenvalue phases. Simultaneous recurrence on their compact torus gives arbitrarily late times when $e^{-itH}$ is arbitrarily close to identity in operator norm. For isometries, $d_{\diamond }({\mathrm{Ad}}_V,{\mathrm{Ad}}_W)\le {\Vert V-W\Vert }_{\infty }$, by referenced pure-state comparison and convexity. Partial trace and Lemma lem:basic give recurrence of the accessible channel and its recovery error. The initial error and universal upper cap both equal $a_d$.