Lemma: Isometry error to complete-channel error

Let $V,W:H_M\to H_B\otimes H_E$ be isometries with $\Vert V-W{\Vert }_{\infty }\le \delta$. Then their full-output channels, and therefore their reduced $B$-channels, satisfy

Proof. Tensoring $V-W$ with a reference identity preserves operator norm. For unit vectors $a,b$,

whose trace norm is at most $2\Vert a-b\Vert$. Apply this to pure referenced inputs, extend by convexity, and divide by two. Partial trace contracts trace distance.