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 Incoming links: Scalar multiple

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 2 / 4 / c / Solution 2026-10-03

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 70 / 5 / a / Solution 2026-10-03

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 15D / b / ii / Solution Created 2026-09-24 Updated 2026-09-25

Inner-product preservation between programs for

U

and

V

gives

⟨ α ∣ β ⟩ ⟨ P_{U} ∣ P_{V} ⟩ = ⟨ α ∣ U^{†} V ∣ β ⟩ ⟨ P_{U} ∣ P_{V} ⟩

for all

∣ α ⟩, ∣ β ⟩

. If

⟨ P_{U} ∣ P_{V} ⟩ \neq = 0

, this identity forces every matrix element of

U^{†} V

to be the same scalar multiple of the corresponding identity matrix element. Hence

U^{†} V = e^{i θ} I

, contrary to physical distinctness. Therefore

⟨ P_{U} ∣ P_{V} ⟩ = 0 .

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