Lemma: Isometric full-output comparison

For isometries $V,\tilde{V}:C\to R\otimes K$,

Consequently both accessible channels and their specified complements are within the same half-diamond allowance.

Proof. For a purified input, also including a reference, the output vectors differ in norm by at most ${\Vert V-\tilde{V}\Vert }_{\infty }$. The trace distance of their pure-state densities is no greater than that vector difference. Convexity covers mixed inputs. Partial trace is a channel and contracts trace distance and diamond distance.