 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-66/1/iv/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

∣ ψ ⟩_{QR} = \sum_{j = 1}^{r} p_{j} ∣ u_{j} ⟩_{Q} ∣ v_{j} ⟩_{R}, p_{j} > 0, \sum_{j} p_{j} = 1.

ρ_{Q} = \sum_{j} p_{j} ∣ u_{j} ⟩ ⟨ u_{j} ∣, ρ_{R} = \sum_{j} p_{j} ∣ v_{j} ⟩ ⟨ v_{j} ∣.

Their nonzero eigenvalues are identical, even if the two ambient dimensions differ. Zero eigenvalues contribute no Von Neumann entropy, so

H (Q)_{ψ} = H (R)_{ψ} = - j \sum p_{j} lo g_{2} p_{j} .

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