Proposition: Two distinct discrimination benchmarks

For computational-basis inputs,

For the states $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$,

Equal-prior optimal single-shot success probabilities are, respectively, $(1+p)/2$ and $(1+p\eta )/2$.

Proof. The common failure block cancels. The remaining key blocks are orthogonal, and Pauli conjugation preserves trace norm. Computational-basis inputs are unchanged by ${\Lambda }_{\eta }$. The difference of the two $X$-eigenstate densities is attenuated by $\eta$. Tensoring ${\sigma }_O$ preserves trace norm. The optimal binary discrimination formula follows by maximizing the effect against the positive part of the difference.