Data-processing inequality for coherent information ($I(A⟩B)-I(A⟩B\prime )=I(A:E|B\prime )\ge 0$)

A quantum channel on the receiving system cannot increase coherent information: $I(A⟩B{)}_{\rho }\ge I(A⟩B^{\prime }{)}_{\sigma }$. Dilate the channel to a linear isometry of Hilbert spaces $B\to B^{\prime }E$. The difference is $I(A:E|B^{\prime })$ of the dilated state, nonnegative by Strong subadditivity of Von Neumann entropy.