Lemma: Validity and exact operational parameters

The operator (eq:pilot-W) is a valid process. Its probabilities are

Its strengths are

and all reverse strengths vanish. The source vector $r$ is acquired from the root laboratory's Pauli outcome expectations.

Proof. Every tensor factor in (eq:pilot-W) is positive. Contracting the initial factor ${{\rho }_r}^T$ with $J({{\mathcal{M}}_{\mathcal{A}}}^a)$ produces ${{\mathcal{M}}_{\mathcal{A}}}^a({\rho }_r)$. Contraction with the depolarizing link, then the identity link, gives the displayed sequential probability. Summing all outcomes gives trace one.

For normalized states,

Trace distance contracts under later channels, so no sender intervention can produce receiver total variation exceeding $p$ across the first link. Replacement preparation of two orthogonal states, followed by their distinguishing measurement, attains $p$. For $A\to C$, let $B$ use the identity. The $B\to C$ link is the identity and permits perfect distinction. Later trace-preserving operations cannot change earlier marginal outcome laws, giving zero reverse strengths.

Measuring the Pauli effects $(I\pm {\sigma }_i)/2$ at $A$, with all later outcomes summed, gives expectation $r_i$.