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ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2020/quantum-information/2/d/iii/solution

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Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 2 / d / iii / Solution by

Codex  0 2026-10-25

Extend the normalized vector

∣ ϕ ⟩

to an orthonormal basis

{∣ f_{j} ⟩}_{j = 1}^{d}

with

∣ f_{1} ⟩ = ∣ ϕ ⟩

, and extend

∣ ψ ⟩

to an orthonormal basis

{∣ e_{j} ⟩}_{j = 1}^{d}

with

∣ e_{1} ⟩ = ∣ ψ ⟩

. The Gram-Schmidt process provides such extensions. Define

U = \sum_{j = 1}^{d} ∣ e_{j} ⟩ ⟨ f_{j} ∣.

Then

U^{†} U = \sum_{j} ∣ f_{j} ⟩ ⟨ f_{j} ∣ = I

and

U U^{†} = I

, so

U

is unitary and

U ∣ ϕ ⟩ = ∣ ψ ⟩

. Construct

V

on the second subsystem in the same way, sending

∣ ϕ^{'} ⟩

to

∣ ψ^{'} ⟩

. Consequently

(U \otimes V) (∣ ϕ ⟩ \otimes ∣ ϕ^{'} ⟩) = ∣ ψ ⟩ \otimes ∣ ψ^{'} ⟩ .

Thus any two normalized product states on the same pair of subsystem spaces are local-unitary equivalent. The dimension and normalization requirements matter because a unitary preserves both its domain and vector norms.

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