Proposition: Allowance bound and an exact small-signal witness

Suppose that complete comparison channels for every intervention at $p$ equal one channel ${\Theta }_q$, and that

uniformly over the declared interventions and contexts. Then the actual signalling strength obeys

A positive allowance does not imply ${\Delta }_D(p,q)=0$.

Proof. The triangle inequality bounds the difference between two actual intervention channels by $2\epsilon$. The half-diamond distance between channels is at most one Watrous (Chapter 3).

For the failure of exactness, on a qubit set

Both maps are channels. Homogeneity of the norm and the input $|1⟩⟨1|$ give

The upper bound is the maximum channel distance one; the input $|1⟩⟨1|$ gives the matching lower bound for $d_{\diamond }(\mathrm{id},{\mathcal{R}}_0)$. The two bit preparations satisfy

The channel is arbitrarily close to a replacement channel while still transmitting an unconditional signal.