Coherent information upper bound by input entropy ($I_c(\Lambda ,\rho )=S(\rho )-I(R:E)\le S(\rho )$)

In a purified Stinespring dilation, coherent information satisfies $I_c=S(RE)-S(E)=S(\rho )-I(R:E)$. Nonnegativity of quantum mutual information implies $I_c\le S(\rho )$. Equality means that the reference and environment are uncorrelated, so the environment learns nothing about the purified input reference.