Prove that $I(A;B|C)=0$ given $I(A;B)=0$ Let $A$, $B$ and $C$ be 3 discrete random variables. If $A$ and $B$ are independent ($I(A;B) = 0$, where $I$ represents the mutual information), how can we prove that $I(A;B|C)=0$? When I draw a Venn diagram this seems trivially true, but I just cannot find a way to prove it.
Your claim is false. Let $A,B,C$ range uniformly over the set
$$
\{(x,y,z) \in \{0,1\}^3 : x \oplus y \oplus z = 0 \}.
$$
On the one hand, $I(A;B) = 0$. On the other hand, $I(A;B|C) = 1$.