Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / b

Table of contents
- Solution b

Solution

For a chosen unit vector

∣ 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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