Theorem: Complementary upper and lower certificates

For a finite qubit channel with complete complement $\mathcal{C}$, put

A polar-isometry decoder satisfies

Every deterministic decoder satisfies

Consequently,

Proof. The Bell-input dilation purifies $J_{\mathcal{C}}$. A Bell pair times a purification of ${\tau }_E$ purifies the product center. Enlarging the decoding ancilla, the polar purification isometry attains root overlap $f\ge 1-{\delta }_E$. The decoded Bell distance is at most $\sqrt{1-f^2}\le \sqrt{2{\delta }_E}$. Lemma lem:choi gives the factor two. The maximally mixed constant decoder has error $3/4$, by the four-Pauli decomposition and the Bell test.

For any decoder, dilate its output and project its pure Bell-input state onto the target Bell vector. Normalizing the environmental projection gives a Bell-product purification at distance $\sqrt{1-F_e}$, when $F_e>0$. After tracing decoder systems, $J_{\mathcal{C}}$ is within that distance of $I_Q/2\otimes \tau$. Since

trace duality gives ${\Vert \mathcal{C}(P)/2\Vert }_1\le 2\sqrt{1-F_e}$. The left side is the complementary pair distance. The Bell-projector test gives error at least $1-F_e$; worst-case overlap cannot exceed Bell overlap. The case $F_e=0$ is immediate.