= Solution
Continue with equal prior probabilities as in part (c). The two <Bloch vectors> are
$$
r=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),\qquad
s=(-\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta).
$$
Their density-matrix difference is $\Delta=\sin\theta\cos\varphi\,\sigma_x$. If a binary measurement effect $E$ declares the first state, its success is
$$
p=\tfrac12\operatorname{Tr}(E\rho_r)+\tfrac12\operatorname{Tr}[(I-E)\rho_s]
=\tfrac12+\tfrac12\operatorname{Tr}(E\Delta).
$$
Diagonalize $\Delta$. Because $0\le E\le I$, the <trace> is maximized by taking $E$ to be its positive-eigenvalue projector. Thus perform a <Pauli measurement> of $\sigma_x$, equivalently measure in $(|0\rangle\pm|1\rangle)/\sqrt2$. If $\cos\varphi>0$, identify the first state on the positive outcome and the second on the negative; reverse the labels if $\cos\varphi<0$. The optimum is
$$
\boxed{p_{\mathrm{correct}}=\frac{1+\sin\theta|\cos\varphi|}{2}.}
$$
For $\cos\varphi=0$ the states coincide, so the best probability is $1/2$. This explicit positive-eigenspace optimization proves the <Helstrom measurement for two pure states> result in the present case, rather than only quoting its name.
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