Solution
= Solution
Expand the difference between the two <quantum mutual informations>:
$$
I(A:BC)-I(A:B)=S(BC)-S(ABC)-S(B)+S(AB).
$$
This is precisely the nonnegative expression in <Strong subadditivity of Von Neumann entropy>. Thus
$$
\boxed{I(A:B)\le I(A:BC).}
$$
The reduced state on $AB$ is obtained by the <partial trace> over $C$, so this proves that discarding a system cannot increase its <quantum mutual information> with the retained reference.