Proposition: Uniform referenced cutoff allowance

If a complete output family satisfies

then, after embedding the original output in the enlarged flag space,

A uniform total-number bound ${\sup }_{\rho }⟨N_{\mathrm{tot}}⟩\le \mu$ implies

For number-block-diagonal output states, the state trace-distance cutoff error equals their discarded probability.

Proof. For a purification with discarded probability $\delta$, direct two-component diagonalization gives

Partial trace is contractive. Adding the overflow block contributes trace norm $\delta$, proving the referenced bound. The same discarded probability is determined by the input marginal, so the uniform assumption holds for reference-entangled inputs. The channel supremum therefore gives (eq:cutoffdiamond).

The operator inequality $N_{\mathrm{tot}}\ge (L+1)(I-P_L)$ proves (eq:cutoffMarkov). For block-diagonal states, removal and overflow insertion occur in orthogonal supports of equal weight, giving exact distance $\delta$.