Proposition: Entropy and occupation underdetermination

Let ${\mathcal{H}}_C\simeq {\mathbb{C}}^m$, $m\ge 2$, and fix a density ${\tau }_T$ with purification $|\chi {⟩}_{T{T}^{\prime }}$. There are complete isometric interfaces whose accessible channels are

For every pure code input, their accessible entropies are equal to $S({\tau }_T)$. Their complete accessible density operators also agree on the selected input $|0⟩$. Every diagnostic on $T$ agrees for all inputs. Nevertheless,

Proof. Complete isometries are

The accessible outputs have the stated form. For pure inputs the $D$ state is pure in both models, so the entropies are $S({\tau }_T)$. The remaining diagnostic assertions follow directly.

Tracing $T$ recovers the good input. The bad accessible channel is constant. For the maximally entangled code-reference input, every decoded bad output has the form $I_Q/m\otimes {\sigma }_C$ and Bell overlap $m^{-2}$. The Bell projection gives the lower bound.