Finite semidefinite certificates

For decoder search, its unnormalized Choi operator $J_D$ must obey

Composition with the known finite channel is linear in $J_D$; the trace-norm objective can be expressed using positive and negative parts as in (eq:complement-SDP). A complete feasible decoder, not only a successful branch, is the required output.

candidate. For a finite channel $\hat{\mathcal{N}}$, a full-rank code state $\tau$, and $P=\mathrm{supp}\hat{\mathcal{N}}(\tau )$, a candidate is

Inverse powers are taken on $P$. The first term is completely positive and has trace $\mathrm{Tr}(PY)$; the completion therefore makes the map CPTP. Writing it down does not prove a small recovery error. Proposition prop:decoder supplies a direct test. This agrees with the distinction emphasized in the supplied Hawk01-B comparison CheungHawk01B.