Theorem: Matrix-unit channel and instrument certificate

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

For complete instruments, replace the matrix-unit norm by

Proof. For a referenced state $\rho ={\sum }_{i,j}|i⟩⟨j|\otimes {\rho }_{ij}$, positivity gives

A purification or positive-block factorization proves this bound. Therefore

The referenced output trace norm is at most $d\epsilon$. For channel differences, maximization over referenced states gives the diamond norm Watrous. For instruments use the channel $\rho \mapsto {⨁}_r{\Phi }_r(\rho )$. Its trace norm is the sum of block trace norms.