Theorem: Finite first-pulse recovery obstruction

For every fixed compact two-pulse model and every selected packet family,

Every decoder also obeys

For every replacement channel on the complete complement,

Proof. The retained-source outputs for the two $X$-eigenstate inputs have an $X$-Bloch separation $2q$. Their distance is at least $q$. The complete complementary distance for that pair is therefore at least $q$.

Lemma lem:leakageBell gives $q\le 2\sqrt{1-F_e}$. Thus $1-F_e\ge q^2/4$. The Bell projector gives $d_{\diamond }(\mathcal{DN},\mathrm{id})\ge 1-F_e$. Worst-case fidelity is no larger than Bell fidelity.

For any fixed replacement output, the triangle inequality for the two complementary inputs gives maximal replacement error at least half their distance, hence at least $q/2$.