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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-323/4/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 323 / 4 / iv / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Choose the trace-distance-preserving binary measurement from the previous part. Let

p, q

be its output probability distributions, so

∥ p - q ∥_{1} = 2 D (ρ, σ)

. The data-processing inequality for quantum relative entropy gives

D (ρ ∥ σ) \geq D (Φ (ρ) ∥Φ (σ)) = D (p ∥ q) .

For diagonal density operators, the quantum expression is the classical Kullback-Leibler divergence. Applying the supplied Pinsker's inequality gives

D (p ∥ q) \geq \frac{1}{2 l n 2} ∥ p - q ∥_{1}^{2} = \frac{2}{l n 2} D (ρ, σ)^{2} .

Consequently the quantum Pinsker inequality is

D (ρ ∥ σ) \geq \frac{2}{ln 2} D (ρ, σ)^{2} .

The

ln 2

factor converts natural-logarithm relative entropy to bits. If the relative entropy is infinite because its support condition fails, the inequality holds automatically.

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