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Helstrom measurement for two pure states

For

ρ_{j} = ∣ α_{j} ⟩ ⟨ α_{j} ∣

, projecting onto the positive and negative eigenspaces of

ρ_{0} - ρ_{1}

attains

P_{s}^{opt} = \frac{1}{2} + \frac{1}{4} ∥ ρ_{0} - ρ_{1} ∥_{1} = \frac{1}{2} (1 + 1 - ∣ ⟨ α_{0} ∣ α_{1} ⟩ ∣^{2}) .

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- Helstrom measurement for the zero and plus states Helstrom measurement for two pure states 19

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