Solution
= Solution
The overlap is
$$
|\langle0|-\rangle|=\frac1{\sqrt2}.
$$
The <Helstrom-Holevo bound> for two equiprobable pure states gives
$$
p_{\rm succ}^{\max}
=\frac12\left(1+\sqrt{1-|\langle0|-\rangle|^2}\right)
=\boxed{\frac12\left(1+\frac1{\sqrt2}\right)}
=\cos^2\frac\pi8.
$$
The <Breidbart basis> is the eigenbasis of the difference of the two weighted state projectors, so its decision rule is precisely the Helstrom measurement and attains this bound.