Proposition: Exact full-code criterion

Let $\mathcal{N}:L({\mathcal{H}}_Q)\to L({\mathcal{H}}_B)$ be a channel with Kraus operators $E_i:{\mathcal{H}}_Q\to {\mathcal{H}}_B$. The following are equivalent:

1. a channel $\mathcal{D}$ satisfies $\mathcal{DN}={\mathrm{id}}_Q$;

2. ${E_i}^{†}{E}_j=c_{ij}{I}_Q$ for scalar coefficients;

3. a complete complementary channel is a replacement channel, $\hat{\mathcal{N}}(X)=\mathrm{Tr}(X)\omega$ for a fixed state $\omega$.

The criterion concerns recovery of the whole input matrix algebra, not merely a subsystem.

Proof. The standard complementary channel has matrix entries

This proves equivalence of (ii) and (iii).

If $K_a$ are Kraus operators of a left inverse, the operators $K_a{E}_i$ form a Kraus family for the identity channel. Its Choi matrix has rank one, so each $K_a{E}_i=c_{ai}{I}_Q$. Trace preservation of $\mathcal{D}$ then gives

Conversely, diagonalize the positive scalar matrix in (ii) by a Kraus-basis change. Remove zero Kraus operators and write

The isometry ranges are orthogonal. Put $\Pi ={\sum }_i{W}_i{W}_i^{†}$ and choose a fixed state $\tau$ on $Q$. Then

is completely positive and trace preserving, and $\mathcal{DN}={\mathrm{id}}_Q$.