Lemma: Normalized Choi comparison

For channels with input dimension $d$,

Proof. The first inequality uses the Bell input. Any pure referenced input can be written $(L\otimes I)|{\Phi }_d⟩$, with $\mathrm{Tr}(L^{†}L)=d$ and ${\Vert L\Vert }_{\infty }^2\le d$. Multiplying the Choi difference on both sides by $L\otimes I$ bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs Watrous (Chapter 3).