Lemma: Normalized Choi conversion

For channels of input dimension $d$,

Proof. A normalized pure referenced input has the form $(L\otimes I)|{\Phi }_d⟩$, with $\mathrm{Tr}(L^{†}L)=d$ and ${\Vert L\Vert }^2\le d$. Multiplying the normalized Choi difference on both sides by $L$ bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs.