The record-only obstruction

Measure-and-prepare channels are entanglement breaking HorodeckiShorRuskai. The quantitative comparison is also stated in CheungHawk04B (Theorem thm:classical). The proof does not prohibit measurement-assisted recovery with a surviving quantum output, as Example ex:assistance shows.

A finite adaptive program whose only returned output is classical has form (eq:record-channel). Indeed, the adjoint image of each classical output projector is an effect $F_x$, and trace preservation makes their sum the identity. Adaptivity and unlimited classical storage do not evade the theorem. An additional input-dependent quantum memory would change its premise.