Theorem: B6: compatible half-diamond error budgets

Suppose an ideal decoder satisfies

and its implementation satisfies

Then

If a finite implementation is a composition of complete channels ${\Phi }_m\cdots {\Phi }_1$, with corresponding implementations ${\tilde{\Phi }}_j$ and local errors $d_{\diamond }({\tilde{\Phi }}_j,{\Phi }_j)\le {\delta }_j$, then

Proof. CPTP pre- and postcomposition contract the diamond distance. Apply the triangle inequality to

For the second statement, telescope the channel products, replacing one factor at a time. Each intervening channel has diamond norm one. The sum of the resulting bounds is the displayed expression.