 Solution

Let ${\rho }_j=|{\alpha }_j⟩⟨{\alpha }_j|$. For a two-outcome measurement with effect $0\le M\le I$ interpreted as guess zero,The positive-eigenspace projector of ${\rho }_0-{\rho }_1$ maximizes the last trace, giving the Helstrom formulaOn the span of the two states, ${\rho }_0-{\rho }_1$ has eigenvaluesThereforeand the positive/negative eigenspace measurement attains equality.

The right side equals one exactly when $|⟨{\alpha }_0|{\alpha }_1⟩|=0$. Thus two pure states are perfectly distinguishable exactly when they are orthogonal.