Proof

Proof. For $q_N<1$, compression and complement measure-and-prepare are completely positive and their traces sum to $\mathrm{Tr}Y$. If $\{e_b\}$ is an orthonormal basis of ${P_N}^{\perp }$, a Kraus family is

The Kraus effects sum to $I_R$. Direct substitution proves the decoded formula. Lemma lem:replacement yields error $q_N{c}_d$. The same conclusion holds for the constant endpoint.

For an arbitrary CPTP decoder, use a Bell input ${\Phi }_d$. The success branch contributes at most $1-q_N$ to decoded Bell overlap. The vacuum branch is independent of the input and has reference marginal $I_Q/d$; its contribution is exactly $q_N/d^2$. The Bell test therefore gives error at least $q_N(1-d^{-2})$.