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

Choose the trace-distance-preserving binary measurement from the previous part. Let $p,q$ be its output probability distributions, so $\Vert p-q{\Vert }_1=2D(\rho ,\sigma )$. The data-processing inequality for quantum relative entropy givesFor diagonal density operators, the quantum expression is the classical Kullback-Leibler divergence. Applying the supplied Pinsker's inequality givesConsequently the quantum Pinsker inequality isThe $\ln 2$ factor converts natural-logarithm relative entropy to bits. If the relative entropy is infinite because its support condition fails, the inequality holds automatically.