Corollary: Reference-stable comparison

Under Theorem thm:comparison, for every $r\ge 1$,

Consequently, equal joint restrictions of two states on ${\mathfrak{M}}_{\mathrm{in}}\otimes M_r$ to ${\mathfrak{A}}_{\mathrm{in}}(I)\otimes M_r$ give equal joint output restrictions on ${\mathfrak{A}}_{\mathrm{out}}(Y)\otimes M_r$.

Proof. Amplify the linear identity (eq:algebrafactor). Complete positivity ensures positivity at every matrix size. Evaluation against the joint states gives the conclusion.