Corollary: Robust detection certificate

If the ideal discrepancy is $\delta$ and the combined state/interface error bound is $e$, then $\delta >e$ certifies

An exactly agreeing actual interface transmits no statistical distinction at any sample size. Ideal agreement alone, followed by implementation error, instead gives only an upper bound on the actual discrepancy; small leakage can still be detected with enough samples.

Proof. The actual discrepancy is at least $\delta -e$ by Theorem thm:robustness and Proposition prop:interfaceerror. Apply the increasing lower-bound function in (eq:chernoff). Exact actual agreement means equality of the output states.