 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-66/3/iv/solution

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

D (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, F (ρ, σ) = Tr ρ σ ρ = ∥ ρ σ ∥_{1} .

The unsquared convention matters: if

P = ∣ ψ ⟩ ⟨ ψ ∣

ρ P ρ

is a rank-one positive operator with its sole nonzero eigenvalue equal to

⟨ ψ ∣ ρ ∣ ψ ⟩

. Hence

F (ρ, P)^{2} = ⟨ ψ ∣ ρ ∣ ψ ⟩ .

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