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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/10d/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 10D / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

A unitary cloner with blank state

∣0 ⟩

would satisfy

U (∣ ψ ⟩ ∣0 ⟩) = ∣ ψ ⟩ ∣ ψ ⟩

(1)

for every

∣ ψ ⟩ \in S

. For

∣ ψ ⟩, ∣ ϕ ⟩ \in S

, preservation of inner products gives

⟨ ψ ∣ ϕ ⟩ = ⟨ ψ ∣ ϕ ⟩^{2} .

(2)

Hence every pair is either orthogonal or represents the same state. Conversely, an orthonormal set can be cloned by defining the map on its basis states and extending it to a unitary. Therefore the no-cloning theorem says that a set of pure states is unitarily clonable exactly when its distinct members are mutually orthogonal.

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