 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/15c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

∣ ψ ⟩

, the two possible outputs have inner product

⟨ ψ ∣ U_{2}^{†} U_{1} ∣ ψ ⟩ .

Part (a) says they can be perfectly distinguished exactly when this inner product is zero. Such an input exists exactly when

0 \in {⟨ ψ ∣ U_{2}^{†} U_{1} ∣ ψ ⟩ : ∥ ψ ∥ = 1} = N (U_{2}^{†} U_{1}) .

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