Proposition: Influence available from an accurate unitary link

If $d_{\diamond }(\mathcal{C},{\mathrm{Ad}}_U)\le ϵ<1/2$ in (eq:comb), then

All reverse strengths vanish. The component order is the three-element chain $A\prec B\prec C$. For the exact one-qubit phase pilot, all three forward strengths equal one.

Proof. At $A$, replace the input by two orthogonal states. Their ideal unitary outputs are perfectly distinguishable. Each actual output is within trace distance $ϵ$ of its ideal counterpart, so their mutual trace distance is at least $1-2ϵ$. A distinguishing binary measurement can be implemented as a receiver instrument. For $A\to C$, let $B$ forward the state.

The identity link from $B$ to $C$ permits perfect distinction. Trace preservation of later operations makes earlier unconditional marginals independent of those operations, proving reverse non-signalling. Positivity of the forward strengths gives the chain.