Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 1 / a / ii

For a purification

∣ Ψ_{ρ} ⟩_{R A}

ρ

F_{e} (ρ, Λ) = ⟨ Ψ_{ρ} ∣ (id_{R} \otimes Λ) (∣ Ψ_{ρ} ⟩ ⟨ Ψ_{ρ} ∣) ∣ Ψ_{ρ} ⟩ .

Λ (X) = \sum_{k} A_{k} X A_{k}^{†}

gives

F_{e} (ρ, Λ) = \sum_{k} ∣ ⟨ Ψ_{ρ} ∣ I \otimes A_{k} ∣ Ψ_{ρ} ⟩ ∣^{2} .

In a Schmidt decomposition of the purification,

⟨ Ψ_{ρ} ∣ I \otimes A_{k} ∣ Ψ_{ρ} ⟩ = \sum_{j} λ_{j} ⟨ j ∣ A_{k} ∣ j ⟩ = Tr (A_{k} ρ) .

Therefore

F_{e} (ρ, Λ) = k \sum ∣ Tr (A_{k} ρ) ∣^{2} .

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