 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/10d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Write

∣ α ⟩ = a ∣0 ⟩ + b ∣1 ⟩

. Expanding in the orthonormal Bell basis of systems

C A

∣ α ⟩_{C} ∣ β_{00} ⟩_{A B} = \frac{1}{2} (∣ β_{00} ⟩_{C A} ∣ α ⟩_{B} + ∣ β_{01} ⟩_{C A} Z ∣ α ⟩_{B} + ∣ β_{10} ⟩_{C A} X ∣ α ⟩_{B} + ∣ β_{11} ⟩_{C A} XZ ∣ α ⟩_{B}) .

Thus, uniformly for

x, z \in {0, 1}

∣ μ_{x z} ⟩ = X^{x} Z^{z} ∣ α ⟩ .

The states are unique because the four Bell states are an orthonormal basis of the two-qubit space. In particular,

∣ μ_{10} ⟩ = X ∣ α ⟩ .

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