= Solution
Choose the <trace-distance-preserving binary measurement> from the previous part. Let $p,q$ be its output probability distributions, so $\lVert p-q\rVert_1=2D(\rho,\sigma)$. The <data-processing inequality for quantum relative entropy> gives
$$
D(\rho\|\sigma)\geq D(\Phi(\rho)\|\Phi(\sigma))=D(p\|q).
$$
For diagonal <density operators>, the quantum expression is the classical <Kullback-Leibler divergence>. Applying the supplied <Pinsker's inequality> gives
$$
D(p\|q)\geq\frac1{2\ln2}\lVert p-q\rVert_1^2
=\frac2{\ln2}D(\rho,\sigma)^2.
$$
Consequently the <quantum Pinsker inequality> is
$$
\boxed{D(\rho\|\sigma)\geq\frac2{\ln2}D(\rho,\sigma)^2}.
$$
The $\ln2$ 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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