Lemma: Bell overlap and half-diamond error

For a channel $\Lambda :C\to C$ of logical dimension $d$, with $F=\mathrm{Tr}(P_d{J}_{\Lambda })$,

Proof. The Bell-projector effect proves the lower bound. For a state $\rho$ and rank-one projector $P$,

This follows by decomposing $\rho$ into pure states, using the pure-state distance formula, trace-norm convexity, and square-root concavity; it is the pure-target Fuchs–van de Graaf bound FuchsVanDeGraaf. Combine it with $d_{\diamond }(\Lambda ,\mathrm{id})\le d{d}_{\mathrm{tr}}(J_{\Lambda },P_d)$, proved in Lemma lem:choicoord.