Lemma: Normalized Choi bounds

For channels $\Lambda ,\mathcal{T}$ with input dimension $d$,

The output dimension need not equal the input dimension.

Proof. The lower bound uses ${\Phi }_d$ as a permissible input. For the upper bound, every pure state on $H\otimes H$ can be written as $(I_H\otimes M)|{\Phi }_d⟩$ with $\mathrm{Tr}(M^{†}M)=d$, and hence ${\Vert M\Vert }_{\infty }\le \sqrt{d}$. For $\Delta =\Lambda -\mathcal{T}$ its output is

The trace norm is at most $d{\Vert J(\Delta )\Vert }_1$. Convexity handles mixed inputs; Schmidt decomposition reduces a larger reference to dimension $d$. Taking the channel-discrimination supremum gives the result.