Theorem: Exact constant-complement criterion

Let $W:C\to A\otimes B$ be a finite-dimensional isometry. The following are equivalent:

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

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

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

3. For a Kraus family $\{A_a\}$ obtained from $W$,

   $${A_a}^{†}{A}_b=c_{ab}{I}_C$$

   for scalar coefficients $c_{ab}$.

Proof. Assume (i), and dilate the decoder by an isometry $T:A\to C\otimes G$. The composite $(T\otimes I_B)W$ dilates the identity channel. The identity channel has rank-one Choi matrix. Since that matrix is a sum of positive rank-one Kraus contributions, every Kraus operator in any such representation is proportional to $I_C$. Consequently

for a fixed unit vector $\chi$. Tracing $CG$ proves (ii).

Conversely, diagonalize ${\tau }_B={\sum }_j{\lambda }_j|j⟩⟨j|$, omitting zero eigenvalues. Write

Applying (ii) to every matrix unit $|i⟩⟨i^{\prime }|$ gives

Let $P_S$ project onto their span and define $U{\varphi }_{ij}=|i{⟩}_C|j{⟩}_J$ on that span. For any fixed density operator ${\sigma }_C$,

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

Finally,

This is a scalar multiple of $\mathrm{Tr}X$ for every $X$ precisely when every ${A_b}^{†}{A}_a$ is scalar. This proves (ii)$\iff$(iii).