Proposition: An exact accumulation witness

On a qubit let

For an unencoded sequence of $k$ such steps,

Consequently, if $k_n\to \infty$, $e_n\to 0$, and $k_n{e}_n\to a>0$, the endpoint error tends to $1-e^{-a}>0$.

Proof. Since ${{\mathcal{R}}_0}^2={\mathcal{R}}_0$,

The half-diamond distance between ${\mathcal{R}}_0$ and identity is one: it is at most one for channels, and the input $|1⟩⟨1|$ gives orthogonal outputs. Homogeneity of the norm gives (eq:noise).

Finally,