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With equal priors, the probability of a correct result from this quantum measurement isThe two required amplitudes areThereforeIts maximum occurs at $\beta =-\pi /8$ and is

Here $|⟨a_0|a_1⟩|=1/\sqrt{2}$, so $\theta =\pi /4$ and $\sin \theta =1/\sqrt{2}$. The stated Helstrom-Holevo bound is consequently exactly the same value, and the basis above is the Helstrom measurement for the zero and plus states.