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ID: past-exam-of-the-introduction-to-quantum-information-course-of-the-university-of-oxford/2021/3/a/solution

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Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 3 / a / Solution by

Codex  0 2026-10-25

The partial trace removes the other orthonormal factor in the Schmidt decomposition:

ρ_{A} = k \sum p_{k} ∣ a_{k} ⟩ ⟨ a_{k} ∣, ρ_{B} = k \sum p_{k} ∣ b_{k} ⟩ ⟨ b_{k} ∣.

(1)

For instance, tracing

∣ a_{k} b_{k} ⟩ ⟨ a_{ℓ} b_{ℓ} ∣

over

B

gives

⟨ b_{ℓ} ∣ b_{k} ⟩ ∣ a_{k} ⟩ ⟨ a_{ℓ} ∣

, which vanishes unless

k = ℓ

. Consequently the nonzero eigenvalues of the two reduced density operators agree, with multiplicity. If the subsystem dimensions differ, the orthonormal Schmidt families can be completed to full bases and the extra eigenvalues are zero.

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