= Mutual-information loss as conditional mutual information
{title2=$I(A:B)-I(A:B\prime)=I(A:E\mid B\prime)\geq0$}
For a channel on $B$, use a <Stinespring dilation> $B\to B'E$. Isometry invariance gives $I(A:B)=I(A:B'E)$, so the loss on discarding $E$ is $I(A:B)-I(A:B')=I(A:E\mid B')$. 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.
Back to article page
