Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 5 / e

For an anti-degradable quantum channel, a channel from $E$ simulates $Q$. The data-processing inequality for quantum relative entropy applied to quantum mutual information gives $I(R:Q)\le I(R:E)$. Part d then implies $I_c(\Lambda ,\rho )\le 0$ for every input. Part c supplies a pure input attaining zero, so