The fixed finite interface

A quantum reference $Q$ is arbitrary and untouched. Every channel equality below is an equality on all operators, not only on a selected preparation ensemble. It therefore remains an equality after tensoring with ${\mathrm{id}}_Q$.

One sufficient preparation/evolution construction is

followed by a complete output regrouping, where $J$ is an isometry, $\Omega$ is a fixed normalized preparation including all necessary purifiers, and $U$ is unitary. Such a construction is not supplied merely by naming its factors.