Proposition: Fixed-table complete-channel approximation

Suppose a second normalized probe vector is installed for each $(a,x)$, with Euclidean distance at most $\delta$ from the first. Keep the same input, output, pinching, and report interfaces. Then

Here $d_{\diamond }(\Phi ,\Psi )=\frac{1}{2}{\Vert \Phi -\Psi \Vert }_{\diamond }$.

Proof. Retain $(a,x)$ as orthogonal flags before common classical postprocessing. The two controlled isometries differ in operator norm by at most $\delta$. For isometries $V,\tilde{V}$, subtraction into two terms gives

Common channel postprocessing contracts the diamond norm, including reference extensions Watrous.