Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 323 / 5 / iii

Let

\overline{ω} = \sum_{j} p_{j} ω_{j}

. When the relative entropies are finite, expanding their trace definitions gives

j \sum p_{j} D (ω_{j} ∥ ρ) - j \sum p_{j} D (ω_{j} ∥ \overline{ω}) = j \sum p_{j} Tr [ω_{j} (lo g_{2} \overline{ω} - lo g_{2} ρ)] = Tr [\overline{ω} (lo g_{2} \overline{ω} - lo g_{2} ρ)] = D (\overline{ω} ∥ ρ) .

j \sum p_{j} D (ω_{j} ∥ ρ) = j \sum p_{j} D (ω_{j} ∥ \overline{ω}) + D (\overline{ω} ∥ ρ) .

For every

p_{j} > 0

ω_{j}

lies in that of

\overline{ω}

, so the first sum on the right is finite in finite dimension. If the support of

\overline{ω}

is not contained in the support of

ρ

, at least one positive-weight

ω_{j}

also violates the support condition. Both sides of the identity are then

+ \infty

. This extends the conclusion without subtracting infinite quantities.

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