Proposition: Finite matrix-unit certification

For two channels on a $d$-dimensional common input, if

then

For complete instruments, replace the norm by the sum of branch norms.

Proof. For a referenced state $\rho ={\sum }_{ij}|i⟩⟨j|\otimes {\rho }_{ij}$, positivity gives $\text{norm}{{\rho }_{ij}}_1\le \sqrt{p_i{p}_j}$, where $p_i=\mathrm{Tr}{\rho }_{ii}$. Hence

The referenced output difference has trace norm at most $d\epsilon$. The channel-difference diamond norm can be optimized over referenced states Watrous. For instruments, use their block-diagonal complete output.