 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2/10d/a/solution

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

For

∣ ψ ⟩ = a ∣0 ⟩ + b ∣1 ⟩

CX (∣ ψ ⟩ ∣0 ⟩) = a ∣00 ⟩ + b ∣11 ⟩ .

A cloned product would instead be

∣ ψ ⟩ ∣ ψ ⟩ = a^{2} ∣00 ⟩ + ab ∣01 ⟩ + ab ∣10 ⟩ + b^{2} ∣11 ⟩ .

Equality up to a physically irrelevant global phase requires

ab = 0

. Thus

CX clones precisely the computational-basis states ∣0 ⟩ and ∣1 ⟩, up to phase .

For a genuine superposition it creates quantum entanglement instead of two independent copies.

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