Proposition: The same receiver survives compatible errors

Suppose proxy and actual complete channels, after access-preserving identifications, differ by at most $\eta$ in half-diamond distance. Suppose the implemented receiver differs from the specified receiver by at most $\iota$. If the specified proxy decoder has half-diamond error $e$, then

A proxy complementary replacement allowance $\zeta$ gives actual leakage at most $\min \{1,\zeta +\eta \}$.

Proof. Partial trace contracts the complete comparison. Insert the same decoder and apply CPTP contraction and the triangle inequality. Do the same with the proxy replacement state.