Proposition: Joint residual information with constant marginals

Let all registers below be qutrits, with arithmetic modulo three. The map

is an isometry. Each one-register output channel is replacement by $I_3/3$. Nevertheless, the joint complement $K_1{K}_2$ contains perfectly distinguishable outputs for distinct computational inputs. In particular, separate constant residual marginals do not certify recovery from $R$.

Proof. The triples in (eq:qutritcode) are distinct for distinct $(r,j)$, proving orthonormality of the encoded basis vectors.

Tracing $K_1{K}_2$ on $|i⟩⟨j|$ forces $r+i=s+j$ and $r+2i=s+2j$, hence $i=j$ and $r=s$. The $R$ output is therefore ${\delta }_{ij}{I}_3/3$. Tracing the other two pairs gives the same identity for $K_1$ and $K_2$.

For input $|j⟩$, the joint complement is supported on

These supports are orthogonal for distinct $j$, since the difference between the two recorded coordinates is $j$. Thus two joint outputs have trace distance one. By the triangle inequality, every replacement state is at distance at least $1/2$ from one of them.

The accessible channel $R$ is constant. On a maximally entangled qutrit input, every decoded output has Bell overlap $1/9$, so its recovery error is at least $8/9$.