= Solution
Choose a <Stinespring dilation> $V:A\longrightarrow B_1E$ of the <quantum channel>. Applying it to the input purification gives the pure vector
$$
|\Omega\rangle_{RB_1E}=(I_R\otimes V)|\psi_{RA}^\rho\rangle.
$$
Its $RB_1$ marginal is the output state in the definition of <coherent information>. Complementary subsystems of a pure state have the same nonzero eigenvalues, by the <Schmidt decomposition>. Thus $S(B_1)=S(RE)$ and $S(RB_1)=S(E)$. It follows that
$$
\boxed{I_c(\Lambda_1,\rho)=S(RE)-S(E)=S(R\mid E).}
$$
Here $S(R\mid E)$ denotes <quantum conditional entropy>, evaluated on the complementary output $RE$. This is <coherent information as an environment conditional entropy>. Equivalently $I_c=-S(R\mid B_1)$; the environment expression has the positive sign, while the receiving-system expression has the negative sign.
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