Proposition: Tensor-free approximate cone certificate

Let $\tilde{T}:\mathfrak{B}\to \mathfrak{M}$ be UCP and let $T_0={\iota }_{I}K:\mathfrak{B}\to \mathfrak{M}$ be a UCP map factoring through $\mathfrak{A}(I)$. If

then states with equal restrictions on $\mathfrak{A}(I)$ have output distance at most

The same estimate holds at every independent reference size when the corresponding joint input restrictions agree.

Proof. The ideal output states agree. Each implemented output differs from its ideal output by at most $e$, because composition of a norm-one state with the difference map has norm at most $2e$. The triangle inequality gives $2e$. State distance is at most one. At matrix size $r$, the difference-map norm is bounded by its completely bounded norm.