Theorem: Decoder contract and fixed later policy

The program defines a deterministic CPTP decoder on all of $R_{k_{\ast }}$. Its optimal value is ${\epsilon }_{\mathrm{rec}}({{\mathcal{N}}_{k_{\ast }}}^U)$. On the success event of Theorem thm:threshold,

For a later cut, let ${\Pi }_{k\to k_{\ast }}$ discard the additional received cells and retain the same early memory. Then the one fixed decoder core with policy

has

for every $k_{\ast }\le k\le K$. Thus the full-diamond error is at most $2e$ at every retained cut.

Proof. The decoder Choi constraints are exactly complete positivity and trace preservation. Composition with the fixed channel is linear in the decoder Choi matrix.

For a Hermiticity-preserving map $\Delta$, the diamond-norm dual is

This follows as the Hermitian specialization of the two-block dual: symmetrizing the two diagonal blocks gives $Z$, and diagonalizing the block matrix gives $Z\pm J(\Delta )\ge 0$. It proves the optimal-value assertion.

Decoder Choi matrices form a compact convex set. The continuous diamond error attains its minimum. The secondary squared-norm minimization is strictly convex and therefore selects a unique decoder matrix. The error estimate follows from Lemma lem:KSW.

Finally, ${\Pi }_{k\to k_{\ast }}{{\mathcal{N}}_k}^U={{\mathcal{N}}_{k_{\ast }}}^U$ by the nested schedule. This proves the fixed-policy identity.