Theorem: Record-only recovery obstruction

Let $\mathcal{M}:C\to Z$ be any channel with a purely classical output, and let $\mathcal{P}:Z\to C$ be any deterministic CPTP decoder. Then

The conclusion concerns record-only processing. It does not apply to an instrument that also retains an authorized quantum output.

Proof. A finite classical-output channel has the form

Only $\mathcal{P}(|z⟩⟨z|)={\rho }_z$ is relevant on this output. Consequently

is measure-and-prepare. Its Bell-input state is separable across $Q\mid C$.

For a normalized product vector $|a{⟩}_Q|b{⟩}_C$,

by Cauchy–Schwarz. Convexity gives the same Bell overlap bound for every separable state. Testing the projector $|{\Phi }_d⟩⟨{\Phi }_d|$ therefore gives trace distance at least $1-d^{-1}$ from the ideal Bell output. This test is one admissible referenced channel test, proving (eq:classical-bound).