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

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 10D / b / Solution by

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

The no-cloning theorem for two pure states states that no single unitary operator can clone two distinct nonorthogonal states. Suppose, to the contrary, that

U ∣ α_{j} ⟩ ∣0 ⟩ = ∣ α_{j} ⟩ ∣ α_{j} ⟩, j = 0, 1.

(1)

Preservation of the inner product by

U

gives

⟨ α_{0} ∣ α_{1} ⟩ = ⟨ α_{0} ∣ α_{1} ⟩^{2} .

(2)

Writing

z = ⟨ α_{0} ∣ α_{1} ⟩

, this requires

z = 0

z = 1

. The first case is orthogonality; the second means that the normalized states represent the same ray. Both contradict the assumptions. Therefore distinct nonorthogonal quantum states cannot be cloned by a unitary process.

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