Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 15C / d / ii

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- Solution ii

Solution

The Born rule says that each probability is the squared norm of the corresponding ancilla component. Since the states are normalized,

∥ ∣ ψ ⟩ ∣ ϕ ⟩ \pm ∣ ϕ ⟩ ∣ ψ ⟩ ∥^{2} = 2 \pm 2∣ ⟨ ψ ∣ ϕ ⟩ ∣^{2} .

Consequently

p_{0} = \frac{1}{2} (1 + ∣ ⟨ ψ ∣ ϕ ⟩ ∣^{2}), p_{1} = \frac{1}{2} (1 - ∣ ⟨ ψ ∣ ϕ ⟩ ∣^{2}) .

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