Lemma: Bell and Pauli witnesses

A decoded constant qubit channel has Bell-input overlap $1/4$. A separable reference–output state has Bell overlap at most $1/2$. For a Pauli channel $\Lambda ={\sum }_P{p}_P{\mathrm{Ad}}_P$,

Proof. A constant channel gives $I_R/2\otimes \sigma$, whose overlap with a maximally entangled qubit state is $1/4$. A product pure state has overlap at most $1/2$; convexity gives the separable bound. For Pauli channels the Bell errors are orthogonal, giving the lower bound $1-p_I$, while the channel triangle inequality gives the same upper bound.