Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 10C / a / Solution

By the Born rule,

P_{ψ} (A, B) = ∣ ⟨ a_{0} b_{0} ∣ ψ ⟩ ∣^{2} + ∣ ⟨ a_{1} b_{1} ∣ ψ ⟩ ∣^{2},

P_{ψ} (B, C) = ∣ ⟨ b_{0} c_{0} ∣ ψ ⟩ ∣^{2} + ∣ ⟨ b_{1} c_{1} ∣ ψ ⟩ ∣^{2},

and

P_{ψ} (C, A) = ∣ ⟨ c_{0} a_{0} ∣ ψ ⟩ ∣^{2} + ∣ ⟨ c_{1} a_{1} ∣ ψ ⟩ ∣^{2} .

Each is the expectation of the projector onto the span of the two displayed matching-label product states.

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