Theorem: Exact complete-complement criterion

For $V:C\to R\otimes B$, the following conditions are equivalent.

1. A deterministic CPTP decoder satisfies $\mathcal{DN}={\mathrm{id}}_C$.

2. There is one state ${\sigma }_B$ such that $\mathcal{E}={\mathcal{T}}_{{\sigma }_B}$.

3. For every logical matrix unit, $\mathcal{E}(E_{ij})={\delta }_{ij}{\sigma }_B$.

4. The normalized complementary Choi state satisfies

   $$J_{\mathcal{E}}=\frac{I_Q}{d}\otimes {\sigma }_B.$$

5. For every residual observable $O\in \mathsf{L}(B)$,

   $$V^{†}(I_R\otimes O)V=\mathrm{Tr}(O{\sigma }_B)I_C.$$

If these conditions hold and $r=\mathrm{rank}{\sigma }_B$, there are a Hilbert space $K$ of dimension $r$, a state ${\sigma }_K$, and an isometry $A:C\otimes K\to R$ such that

An exact decoder on all of $\mathsf{L}(R)$ is

Every external reference is preserved.

Proof. Matrix units span $\mathsf{L}(C)$, so (ii) and (iii) are equivalent. Expansion of (eq:choi-definition) gives equivalence with (iv). Trace duality gives

which proves equivalence with (v).

Assume (ii), and diagonalize

The residual support condition implies

The coefficients of $\mathcal{E}(E_{ij})$ give

Thus the rule $A(|i⟩\otimes |\alpha ⟩)=|r_{i\alpha }⟩$ defines an isometry, and ${\sigma }_K={\sum }_{\alpha }{p}_{\alpha }|\alpha ⟩⟨\alpha |$ gives (eq:normal-form). The two terms in (eq:exact-decoder) are completely positive; their traces sum to $\mathrm{Tr}Y$. Substitution of (eq:normal-form) proves (i).

Conversely, assume (i). A finite Stinespring dilation of the decoder has an isometry $L:R\to C\otimes G$ Stinespring. Define

The decoded output on input $|i⟩$ is pure, hence

for normalized ${\eta }_i$. Exact recovery of $E_{ij}$ implies $⟨{\eta }_j\mid {\eta }_i⟩=1$. All ${\eta }_i$ therefore equal one vector $\eta$. Tracing $C$ and $G$ gives

which is (iii).

Equality of channels on all logical matrix units persists after tensoring with any reference identity. This proves reference preservation.