Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 3 / ii / b

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- Solution b

Solution

Because the two flags

E_{C}

and

F_{C}

are orthogonal projections, the flagged state is block diagonal. If

h (λ)

denotes the binary entropy, then

S (ρ_{A BC}) = h (λ) + λ S (ρ_{A B}^{'}) + (1 - λ) S (ρ_{A B}^{''}),

and

S (ρ_{BC}) = h (λ) + λ S (ρ_{B}^{'}) + (1 - λ) S (ρ_{B}^{''}) .

Its unflagged marginals are

ρ_{A B} = λ ρ_{A B}^{'} + (1 - λ) ρ_{A B}^{''}

and

ρ_{B} = λ ρ_{B}^{'} + (1 - λ) ρ_{B}^{''}

. Substitution in Strong subadditivity of Von Neumann entropy cancels the two binary-entropy terms and gives

H (A ∣ B)_{λ ρ^{'} + (1 - λ) ρ^{''}} \geq λ H (A ∣ B)_{ρ^{'}} + (1 - λ) H (A ∣ B)_{ρ^{''}} .

This is exactly the concavity of quantum conditional entropy.

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