Proposition: Vanishing local error can remain macroscopic

Let

For $p_m=\alpha /m$,

Although each stage has half-diamond allowance at most $\alpha /m$,

for $D\ge 2$ and $\alpha >0$.

Proof. Use ${{\mathcal{R}}_{\mathcal{D}}}^2={\mathcal{R}}_D$ to obtain the composition formula. Apply the channel to half of a maximally entangled state and test the maximally entangled projector. The replacement output has overlap $D^{-2}$. Finally $(1-\alpha /m{)}^m\to e^{-\alpha }$.