Theorem: Exact recovery criterion

Let $V:{\mathcal{H}}_A\to {\mathcal{H}}_Z\otimes {\mathcal{H}}_F$ be a finite-dimensional isometry with accessible channel $\mathcal{N}$ and complementary channel $\mathcal{C}$. The following are equivalent:

1. a CPTP decoder satisfies $\mathcal{DN}={\mathrm{id}}_A$;

2. there is a state ${\tau }_F$ such that

   $$\mathcal{C}(X)=\mathrm{Tr}(X){\tau }_F\text{for all }X\in \mathcal{B}({\mathcal{H}}_A);$$

3. for a Kraus family of $\mathcal{N}$,

   $${A_a}^{†}{A}_b=c_{ab}{I}_A$$

   with scalar coefficients $c_{ab}$.

Proof. Suppose a decoder exists and let $T$ be its Stinespring isometry. The composite $(T\otimes I_F)V$ dilates the identity channel. Its Kraus operators are all proportional to the identity: the Choi matrix of the identity is rank one, and a sum of positive rank-one Choi contributions can have that range only when every contribution lies in it. Thus

for a fixed ancillary vector $\chi$. Tracing all but $F$ proves (eq:constantcomplement).

Conversely, diagonalize ${\tau }_F={\sum }_j{\lambda }_j|j⟩⟨j|$, omitting zero eigenvalues, and write

Applying (eq:constantcomplement) to every matrix unit gives

On their span $S$, define the isometry $U|{\varphi }_{ij}⟩=|i{⟩}_A|j{⟩}_J$. For a fixed state ${\sigma }_A$,

is CPTP and returns $X$ when applied to $\mathcal{N}(X)$. Finally,

so constancy on every matrix unit is equivalent to (eq:KL).