= Solution
Implement the <quantum channel> by a unitary dilation with a pure initial environment. The final state on $RBE$ is pure. The <Schmidt decomposition> across either bipartition gives equal complementary <Von Neumann entropies>, so $S(RB)=S(E)$ and $S(B)=S(RE)$. By the <Subadditivity of Von Neumann entropy>,
$$
I_c(\Phi,\rho)=S(RE)-S(E)\leq S(R).
$$
The reference is untouched and purifies the input, hence $S(R)=S(\rho)$. Therefore \b[$\boxed{I_c(\Phi,\rho)\leq S(\rho)}$.]
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