Coherent information as an environment conditional entropy ($I_c(\Lambda ,\rho )=S(R\mid E)=-S(R\mid B)$)

Purify the input on $RA$ and dilate a channel by $A\to BE$. The joint state $RBE$ is pure, so complementary entropies obey $S(B)=S(RE)$ and $S(RB)=S(E)$. Therefore coherent information is $I_c=S(B)-S(RB)=S(R\mid E)=-S(R\mid B)$. The positive environment conditional entropy and negative receiving-system conditional entropy are two equivalent forms.