 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-48/1/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Use logarithms to base two, so all Von Neumann entropies and Shannon entropies are measured in bits. Write the nonzero eigenvalues of

ρ_{i}

λ_{ij}

p_{i} λ_{ij}

. Therefore

S (ρ) = - i, j \sum p_{i} λ_{ij} lo g_{2} (p_{i} λ_{ij}) = - i \sum p_{i} lo g_{2} p_{i} j \sum λ_{ij} - i \sum p_{i} j \sum λ_{ij} lo g_{2} λ_{ij} = H (p) + i \sum p_{i} S (ρ_{i}) .

Each

ρ_{i}

has trace one. Terms with zero probability or zero eigenvalue vanish under the convention

0 lo g 0 = 0

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