Theorem: Complete operational discrimination

For two named message preparations,

All ordinary and failure records are included.

Applying $\mathcal{D}$ and then $Z$-readout gives ideal gap $p$ for $|0⟩,|1⟩$. Applying $\mathcal{D}$ and then $X$-readout gives ideal gap $p\eta$ for $|+⟩,|-⟩$.

Proof. An effect expectation changes by at most the trace distance of the complete records. Apply this to each preparation and use the reverse triangle inequality. The two explicit gaps follow from $\mathcal{D}{\mathcal{N}}_2=p{\Lambda }_{\eta }+(1-p)\mathcal{T}$.