Proposition: Finite autonomous recurrence

Let $H$ be a fixed finite-dimensional Hamiltonian and let $V_t=e^{-itH}{V}_0$ describe a fixed prepared interface. There are arbitrarily late times with complete-channel distance arbitrarily close to that at $t=0$. If the initial accessible channel is replacement, then

Proof. There are finitely many eigenvalues. Simultaneous recurrence of their phases gives arbitrarily late times for which $e^{-itH}$ is arbitrarily close to the identity in operator norm. For isometries,

Indeed, apply the pure-state trace-distance bound to every referenced input and then use convexity. Partial trace gives accessible-channel recurrence. Lemma lem:stability gives recovery-error recurrence. The initial error and the universal upper bound are both $a_d$.