Helstrom-Holevo bound

Equiprobable pure states of overlap magnitude $c$ have optimal discrimination probability $\frac{1}{2}(1+\sqrt{1-c^2})$.

For ${\rho }_j=|{\alpha }_j⟩⟨{\alpha }_j|$, projecting onto the positive and negative eigenspaces of ${\rho }_0-{\rho }_1$ attains

For equiprobable states $|0⟩$ andthe orthonormal measurement basis at angle $\beta =-\pi /8$ to the computational basis attains

Two pure states can be distinguished with certainty in one measurement exactly when their inner product is zero.

Unambiguous quantum state discrimination allows an inconclusive outcome but forbids a wrong conclusive identification. Two distinct pure states admit such a measurement, whereas two full-rank density operators on the same support do not.