Proposition: A sufficient relay-gap certificate

Suppose an ideal discrete relay pair has laws ${\delta }_0,{\delta }_1$. If its two implemented laws are each within total variation ${\eta }_{\epsilon }$ of the corresponding ideal law, then

For a finite composition of complete channels with local half-diamond errors ${\delta }_{\epsilon ,j}$, one may take ${\eta }_{\epsilon }\le {\sum }_j{\delta }_{\epsilon ,j}$.

Proof. The first claim is the triangle inequality. For the second, telescope the channel compositions one factor at a time. CPTP pre- and postcomposition contract diamond distance. The resulting total channel error is at most the sum of local errors; a terminal readout cannot increase it.