Theorem: Complete-complement criterion

For (eq:isometry)–(eq:channels), the following are equivalent:

1. there is a CPTP decoder $D$ with $D\mathcal{N}={\mathrm{id}}_C$;

2. there is a density operator ${\tau }_B$ such that

   $$\mathcal{C}(X)=\mathrm{Tr}(X){\tau }_B\text{for every }X\in \mathcal{B}(C);$$

3. in one, hence every, orthonormal code basis,

   $$\mathcal{C}(|i⟩⟨j|)={\delta }_{ij}{\tau }_B;$$

4. for the Kraus family $K_a=(I_A\otimes ⟨a|)V$,

   $${K_a}^{†}{K}_b=c_{ab}{I}_C$$

   for scalar coefficients $c_{ab}$.

If these conditions hold and $r=\mathrm{rank}{\tau }_B$, then

The recovered channel preserves every external reference.

Proof. Suppose $D\mathcal{N}={\mathrm{id}}_C$, and dilate $D$ by an isometry $W:A\to C\otimes G$. Then

dilates the identity channel. The Choi matrix of identity has rank one. Every Kraus operator of $T$, whose vectorizations give positive summands of that matrix, is therefore proportional to $I_C$. Consequently

for a single unit vector $\chi$. Tracing $CG$ proves (eq:constantcomplement).

Conversely, diagonalize ${\tau }_B={{\sum }_{a=1}}^r{\lambda }_a|a⟩⟨a|$, with ${\lambda }_a>0$. The isometry has no component in the kernel of ${\tau }_B$. Write

The complementary matrix-unit identities give

Thus $A$ contains $dr$ orthonormal vectors, proving (eq:rankcapacity).

Let $P$ project onto their span, and define the isometry $U:PA\to C\otimes {\mathbb{C}}^r$ by $U{u}_{ia}=|i⟩|a⟩$. For any fixed code state $\sigma$,

is CPTP and satisfies $D\mathcal{N}(X)=X$.

Matrix units span $\mathcal{B}(C)$, proving the equivalence of conditions 2 and 3. Finally,

so constancy is equivalent to condition 4. Equality of channels remains equality after tensoring with any identity map, proving reference preservation.