Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 4 / c / v / Solution

Combining parts (iii) and (iv), then multiplying by

1 + ε

, gives

H (A ∣ B)_{σ} - H (A ∣ B)_{ρ} \leq ε (H (A ∣ B)_{Δ^{'}} - H (A ∣ B)_{Δ}) + (1 + ε) H (\frac{ε}{1 + ε}) .

The dimension bound for quantum conditional entropy is

- lo g d_{A} \leq H (A ∣ B)_{τ} \leq lo g d_{A} .

Indeed, Subadditivity of Von Neumann entropy gives

H (A ∣ B)_{τ} \leq S (τ_{A}) \leq lo g d_{A}

, while the Araki–Lieb inequality gives

H (A ∣ B)_{τ} \geq - S (τ_{A}) \geq - lo g d_{A}

. Hence

H (A ∣ B)_{σ} - H (A ∣ B)_{ρ} \leq 2 ε lo g d_{A} + (1 + ε) H (\frac{ε}{1 + ε}) .

Interchanging

ρ

and

σ

proves the continuity bound for quantum conditional entropy:

∣ H (A ∣ B)_{ρ} - H (A ∣ B)_{σ} ∣ \leq 2 ε lo g d_{A} + (1 + ε) H (\frac{ε}{1 + ε}) .

Solved by gpt-5.6-sol high.

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