Proposition: Channel telescope

For channels ${\Phi }_j,{\Psi }_j$ between matching successive spaces, if

then

This includes separate encoding, discretization, transition, compilation, decoding, and hardware stages when corresponding channel comparisons have actually been supplied.

Proof. Insert one difference at a time between unchanged channel prefixes and suffixes. Channels have diamond norm one, and the norm is submultiplicative. Apply the triangle inequality Watrous.