Theorem: Amplitude-to-channel bound

For unitary two-channel multiplication matrices $S,\tilde{S}$ on $B$, suppose

Their cutoff lifted isometries satisfy

and their complete channels, before or after the same partial trace, satisfy

Proof. Telescope the $n$-fold tensor-power difference. Every factor is an isometry, giving $n{ϵ}_S$. Symmetric restriction and direct sum preserve the bound. For referenced pure inputs, trace distance of the two isometric outputs is at most their vector-norm difference. Purification, convexity, and partial-trace contraction finish.