Theorem: Referenced packet recovery upper bound

For the comparison channel ${\mathcal{N}}_0$, define

Then

Consequently

A useful explicit upper estimate is

Proof. Theorem thm:source gives the final source $Y$-mean

For every input and every reference extension, the probability outside $|\chi ⟩$ is therefore at most $p_{\lambda }$.

Let ${\zeta }_{RSh}$ be the actual joint source-packet output. Lemma lem:reset compares it with $|\chi ⟩⟨\chi |\otimes {\zeta }_{Rh}$. Apply ${U_0}^{†}$ on $Sh$, retaining $R$, and trace the packet. The first state returns the input exactly; the second gives $({\mathrm{id}}_R\otimes {\mathcal{D}}_{\lambda }{\mathcal{N}}_0)(\rho )$. Contractivity proves the uniform referenced bound.

Since $1-e^{-x}\le x$,

This proves the explicit estimate and convergence. An effect expectation changes by at most trace distance, giving both fidelity bounds.