Proposition: Arbitrarily late recurrence of a fixed finite proxy

Let $U$ be a unitary on a finite-dimensional space. There are positive integers $m_k\to \infty$ such that ${\Vert U^{m_k}-I\Vert }_{\infty }\to 0$. Consequently, for a fixed code preparation and fixed output factorization, the corresponding complete and reduced channels return arbitrarily close to their time-zero channels along a subsequence.

Proof. The unitary group in finite dimensions is compact. Thus some subsequence $U^{n_k}$ converges. Pass to a further subsequence with $n_{k+1}-n_k\ge k$, and set $m_k=n_{k+1}-n_k$. Then

while $m_k\to \infty$. Multiplying by the fixed preparation gives convergence of the complete isometries in operator norm. The isometry-to-channel estimate in the proof of Theorem thm:endpoint, followed by partial-trace contraction, gives the channel conclusion.