Mutual-information loss as conditional mutual information ($I(A:B)-I(A:B\prime )=I(A:E\mid B\prime )\ge 0$)

For a channel on $B$, use a Stinespring dilation $B\to B^{\prime }E$. Isometry invariance gives $I(A:B)=I(A:B^{\prime }E)$, so the loss on discarding $E$ is $I(A:B)-I(A:B^{\prime })=I(A:E\mid B^{\prime })$. This quantum conditional mutual information is nonnegative by Strong subadditivity of Von Neumann entropy. The identity holds for arbitrary mixed inputs and identifies the correlations lost to the discarded environment.