Proposition: Finite composition error

For compatible channels define

If the implemented preparation, readout, $m$ route gates, and output interface differ from their ideal versions by at most ${\epsilon }_{\mathrm{prep}}$, ${\epsilon }_{\mathrm{read}}$, ${\epsilon }_1,\dots ,{\epsilon }_m$, and ${\epsilon }_{\mathrm{out}}$, respectively, then the complete output channel differs by at most

Proof. Insert and subtract hybrid compositions in which one component at a time is replaced by its ideal version. The triangle inequality and contraction of the diamond norm under pre- and postcomposition by channels give the bound.