Theorem: Classical-output and measure-and-prepare obstruction

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

In particular, the bound applies to every finite adaptive process whose only retained output is classical.

Computational readout followed by basis-state preparation attains the bound.

Proof. The Bell-input output of $\mathcal{D}\mathcal{E}$ is separable across the reference–output partition. For a normalized product vector,

by Cauchy–Schwarz. Convexity gives Bell overlap at most $1/d$ for every separable state. Testing the Bell projector proves (eq:classical-obstruction).

A channel to a finite classical register has the form

A decoder from that register prepares a state for each outcome. Adaptive processing before the final classical output does not change this form.

For attainability, define

Computational dephasing is

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