Mutual information and coherent information identity ($I(R:B)=S(\rho )+I_c(\Lambda ,\rho )$)

For a purified input with reference $R$ and channel output $B$, $I(R:B)=S(R)+S(B)-S(RB)=S(\rho )+I_c(\Lambda ,\rho )$. The reference entropy is fixed through subsequent channels. Thus data-processing inequality for coherent information immediately implies data processing for quantum mutual information between that reference and successive outputs.