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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/10d/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 10D / c / Solution by

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

Suppose first that the stated unitary and environment states exist. Preservation of the inner product gives

⟨ ϕ_{0} ∣ ϕ_{1} ⟩ = ⟨ ψ_{0} ∣ ψ_{1} ⟩ ⟨ e_{0} ∣ e_{1} ⟩ .

The environment states are normalized, so the Cauchy-Schwarz inequality gives

∣ ⟨ e_{0} ∣ e_{1} ⟩ ∣ \leq 1

. Therefore

∣ ⟨ ϕ_{0} ∣ ϕ_{1} ⟩ ∣ \leq ∣ ⟨ ψ_{0} ∣ ψ_{1} ⟩ ∣.

Conversely, write

a = ⟨ ϕ_{0} ∣ ϕ_{1} ⟩, b = ⟨ ψ_{0} ∣ ψ_{1} ⟩

and assume

∣ a ∣ \leq ∣ b ∣

. If

b \neq = 0

, put

c = a / b

, so

∣ c ∣ \leq 1

, and choose

∣ e_{0} ⟩ = ∣0 ⟩, ∣ e_{1} ⟩ = c ∣0 ⟩ + 1 - ∣ c ∣^{2} ∣1 ⟩ .

Then

⟨ e_{0} ∣ e_{1} ⟩ = c

and the desired output vectors have inner product

b c = a

, exactly matching the input vectors. If

b = 0

, the inequality forces

a = 0

, and one may take

∣ e_{0} ⟩ = ∣ e_{1} ⟩ = ∣0 ⟩

.

In either case the input pair and output pair have the same Gram matrix. The isometry taking one pair to the other extends to a unitary matrix, so the required

U

exists. Hence

U exists ⟺ ∣ ⟨ ϕ_{0} ∣ ϕ_{1} ⟩ ∣ \leq ∣ ⟨ ψ_{0} ∣ ψ_{1} ⟩ ∣,

which is the environment-assisted two-state pure-state transformation criterion.

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