Theorem: Exact complete-complement criterion

Let $W:C_d\to R\otimes B$ be a finite-dimensional isometry and let $\mathcal{N},\mathcal{C}$ be its two reductions. The following are equivalent:

1. a CPTP decoder $D:\mathcal{B}(R)\to \mathcal{B}(C_d)$ satisfies $D\mathcal{N}={\mathrm{id}}_{C_d}$;

2. $\mathcal{C}(M)=\mathrm{Tr}(M){\tau }_B$ for one state ${\tau }_B$;

3. $\mathcal{C}(E_{ij})={\delta }_{ij}{\tau }_B$ for every matrix unit.

If $r=\mathrm{rank}{\tau }_B$, then $\dim R\ge dr$. The recovered channel preserves every independent external reference.

Proof. The equivalence of the last two statements is linearity. For the forward direction, dilate the decoder by an isometry $L:R\to C_d\otimes G$. Then

dilates the identity. Purity of its decoded marginal on a basis input gives $Z|i⟩=|i⟩|{\chi }_i⟩$. Recovery of $E_{ij}$ gives $⟨{\chi }_j,{\chi }_i⟩=1$. All normalized ${\chi }_i$ coincide. Tracing $C_{d}G$ proves constancy of the complement.

Conversely, diagonalize ${\tau }_B={{\sum }_{a=1}}^r{\lambda }_a|a⟩⟨a|$, with ${\lambda }_a>0$. Write

The complementary matrix-unit identities give

Thus $dr$ vectors in $R$ are orthonormal. Define the isometry $T:C_d\otimes {\mathbb{C}}^r\to R$ by $T|i,a⟩=|u_{ia}⟩$, and let ${\tau }_K={\sum }_a{\lambda }_a|a⟩⟨a|$. Then

For any fixed state $\omega$ on $C_d$,

is CP and trace preserving on all of $R$, and recovers every matrix unit. Equality of channels remains equality after reference amplification.