Theorem: Classical-output obstruction, locally restated

Let $\mathcal{E}$ be a measure-and-prepare channel from $C$, and let $\mathcal{D}$ be any CPTP decoder to $C$. Then

The bound applies to every finite adaptive process whose only retained output is classical. Computational readout followed by basis-state preparation attains it.

Proof. A measure-and-prepare channel has the form

Its Bell-input output, and any locally decoded version, is separable across reference and output HorodeckiShorRuskai. For normalized product vectors,

Convexity gives Bell overlap at most $d^{-1}$ for every separable state. Testing the target Bell projector proves (eq:classical).

A channel to a finite classical register has POVM form. Any decoder from it prepares an output state for each record. Finite adaptive processing before the final record does not change this measure-and-prepare form.

For attainability, let

Computational dephasing is

These $d$ unitary channels have orthogonal Bell outputs. Lemma lem:unitary-mixture gives $d_{\diamond }({\Delta }_d,{\mathrm{id}}_C)=1-d^{-1}$.