Theorem: Deterministic decoder certificate

Program (eq:SDP) specifies a decoder on the whole accessible space, including inputs outside the encoded output support. Its optimal value is ${\epsilon }_{\mathrm{rec}}({{\mathcal{N}}_k}^U)$. On the event of Theorem thm:threshold,

Equivalently, the full diamond norm is at most $2\min \{a_d,\sqrt{2\zeta }\}$.

Proof. The Choi constraints are precisely complete positivity and trace preservation. Composition with the fixed channel is linear in $C_{\mathcal{D}}$.

For a Hermiticity-preserving map, the finite diamond-norm dual is

Applying it to the decoded difference gives (eq:SDP).

The upper-bound direction can also be checked directly: $Z\pm J(\Delta )\ge 0$ define two CP maps whose difference is $\Delta$, while their sum has Choi matrix $Z$. On any referenced state, its output trace norm is at most the trace of that CP sum, hence at most $t$.

Finite decoder Choi matrices form a compact set. Diamond error is continuous, so an optimum is attained. A norm-dual variable is attained as well; bounded $t$ bounds $\mathrm{Tr}Z$, and $Z\ge 0$, so a minimizing sequence has a convergent subsequence. Strict convexity of the Frobenius norm makes the stated secondary choice unique on the convex optimal decoder set.

Finally, Lemma lem:approx bounds this optimum on the decoupling event.