Proposition: Composition and state-error budgets

For composable regulated CPTP channels ${\Phi }_i$ and ${\tilde{\Phi }}_i$,

Consequently, if the complete physical and modeled ensembles have a certified total average trace-distance allowance

where each term is defined on the relevant common state/channel interface, then the score changes by at most ${ϵ}_{\mathrm{tot}}$.

Proof. Expand the difference of the channel compositions as a telescoping sum with one factor difference in each term. CPTP pre- and post-composition do not increase the diamond norm. The triangle inequality proves (eq:telescoping). For a mixture of state and channel allowances, insert the intermediate output ensembles and use trace-distance contractivity and the triangle inequality. Apply Proposition prop:scorecontinuity.