 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2021/quantum-information/1/b/i/solution

No. Perfect discrimination of pure states requires orthogonality, whereas

⟨ ϕ_{0} ∣ ϕ_{1} ⟩ = \frac{1}{4} - \frac{1}{4} + \frac{1}{2} = \frac{1}{2} \neq = 0.

To see the obstruction directly for a basic measurement, let

{∣ b_{j} ⟩}

be its orthonormal basis. Certain identification would require disjoint sets of possible outcome labels for two different preparations. For every

j

, at least one of

⟨ b_{j} ∣ ϕ_{0} ⟩

and

⟨ b_{j} ∣ ϕ_{1} ⟩

would therefore vanish. Completeness would then give

⟨ ϕ_{0} ∣ ϕ_{1} ⟩ = \sum_{j} ⟨ ϕ_{0} ∣ b_{j} ⟩ ⟨ b_{j} ∣ ϕ_{1} ⟩ = 0,

contradicting the computed overlap. Failure to distinguish just this pair already rules out certain identification of the three-state family.

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