Lemma: Choi and matrix-unit conversions

For channels with common input dimension $D$,

If all matrix-unit trace-norm differences are at most $r$, then $d_{\diamond }(\Phi ,\Psi )\le \min (1,Dr/2)$.

Proof. A pure referenced input is $(A\otimes I)|{\Phi }_D⟩$, up to a reference isometry, with ${\Vert A\Vert }^2\le D$. Trace-norm multiplication proves the Choi bound. Alternatively write its vector as ${\sum }_{\alpha }{v}_{\alpha }\otimes |\alpha ⟩$, with ${\sum }_{\alpha }{\Vert v_{\alpha }\Vert }^2=1$. The output difference norm is at most $r({\sum }_{\alpha }\Vert v_{\alpha }\Vert {)}^2\le Dr$. Divide by two and use convexity.