Proposition: Classical-record reconstruction obstruction

Let $Q$ be a $d$-dimensional quantum input, $d\ge 2$. Every channel from $Q$ to a finite classical record followed by a channel from that record to a quantum output is entanglement breaking. It cannot equal ${\mathrm{id}}_Q$, including on reference-entangled inputs. Its half-diamond distance from ${\mathrm{id}}_Q$ is at least $1-1/d$.

Proof. The composite has measure-and-prepare form

For an arbitrary reference, its output is a sum of positive product operators and is therefore separable.

Apply the channel to one half of $|{\Phi }_d⟩=d^{-1/2}{{\sum }_{j=1}}^d|j⟩|j⟩$. Every separable state has overlap at most $1/d$ with $|{\Phi }_d⟩⟨{\Phi }_d|$: this follows for pure product states from Cauchy–Schwarz and then for separable mixtures by convexity. Testing the Bell projector gives trace distance at least $1-1/d$ from the ideal output. The half-diamond distance dominates this particular referenced-input distance.