Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 4 / e

Table of contents
- Solution e

Solution

For outcome probabilities

p_{k} = Tr (Λ_{k} ρ)

and

q_{k} = Tr (Λ_{k} σ)

, the hint gives

\frac{1}{2} \sum_{k} ∣ p_{k} - q_{k} ∣ \geq \frac{1}{2} \sum_{k} (p_{k} - q_{k})^{2} = 1 - \sum_{k} p_{k} q_{k} .

The left side is at most

∥ ρ - σ ∥_{1} /2

by part d. Choose a POVM attaining the stated minimum characterization of quantum fidelity; then

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

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