= Helstrom measurement for two pure states
{c}
Suppose a preparation chooses normalized <pure states> $|\alpha_0\rangle,|\alpha_1\rangle$ with equal <prior probabilities>. Write $\rho_j=|\alpha_j\rangle\langle\alpha_j|$. The optimal minimum-error <quantum measurement> assigns the positive and negative eigenspaces of $\rho_0-\rho_1$ to guesses 0 and 1, respectively. Its success <probability> is
$$
P_s^{\rm opt}=\frac12+\frac14\|\rho_0-\rho_1\|_1
=\frac12\left(1+\sqrt{1-|\langle\alpha_0|\alpha_1\rangle|^2}\right).
$$
Zero-eigenvalue directions may be assigned arbitrarily. Orthogonal <quantum states> give certainty, while identical <quantum states> allow only <probability> $1/2$. This is minimum-error discrimination; it does not introduce an inconclusive outcome.
For unequal <prior probabilities> $p_0,p_1$ with $p_0+p_1=1$, use the positive and negative eigenspaces of $p_0\rho_0-p_1\rho_1$ instead. For two <pure states>,
$$
P_s^{\rm opt}=\frac12\left(1+\sqrt{1-4p_0p_1|\langle\alpha_0|\alpha_1\rangle|^2}\right).
$$
Thus identical states reduce to guessing the more likely preparation. Omitting the prior weights would generally select the wrong measurement. <The Theory of Quantum Information by John Watrous>[Section 3.1.1, Theorem 3.4, Holevo–Helstrom theorem with prior parameter lambda]{ref}
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