Proposition: Same fixed decoder under compatible transfer

Suppose actual and proxy complete channels are compared on the same code after access-preserving identifications, with distance at most $ϵ$. If a proxy decoder has error $e$, and its implemented accessible decoder differs from it by at most $\iota$, then

A proxy complementary replacement allowance $\zeta$ gives

Proof. Partial trace contracts the complete comparison. Insert the same fixed decoder and use contraction and the triangle inequality. Do the same for the proxy replacement state.