Proposition: Deterministic receiver and fixed later policy

Program (eq:sdp) defines a unique selected CPTP receiver. Its objective value is ${\epsilon }_{\mathrm{rec}}({\mathcal{N}}_{\ast }^U)$. For $k\ge k_{\ast }$, define

where ${\Pi }_{k\to k_{\ast }}$ discards the subsequently received cells. Then

Thus one decoder core supplies all later retained certificates.

Proof. For a Hermiticity-preserving map $\Delta$, diamond-norm duality gives

The Choi constraints on $C$ are exactly CPTP, and composition is linear in $C$, proving the objective assertion.

Decoder Choi matrices form a compact convex set; the continuous error attains its minimum. The minimizing decoder set is convex, and the strictly convex secondary norm selects a unique matrix. Finally ${\Pi }_{k\to k_{\ast }}{{\mathcal{N}}_k}^U={\mathcal{N}}_{\ast }^U$ by the stipulated schedule.