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 $I(R:Q)\leq I(R:E)$. Part d then implies $I_c(\Lambda,\rho)\leq0$ for every input. Part c supplies a pure input attaining zero, so
$$
\boxed{Q^{(1)}(\Lambda)=\max_\rho I_c(\Lambda,\rho)=0}.
$$

Solution

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

{wiki=Quantum_channel#Reversible_channel}

A quantum channel is anti-degradable when its receiver output can be simulated from the complementary environment output. Data processing then forces every coherent information to be nonpositive, and no-cloning prevents perfect quantum transmission.


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