Mutual information quantifies to what degree $X$ decreases the uncertainty about $Y$. However, to my understanding, it does not quantify "in how many ways" $X$ decreases the uncertainty. E.g., consider the case where $X$ is a 3D vector, and consider $X_1=[Y,0,0]$ vs. $X_2 = [Y,Y^2, 3.5Y]$. Intuitively, $X_2$ contains "more information" about $Y$, or is more redundant with respect to $Y$, than $X_1$; but if I understand correctly, both have the same mutual information. Is there an alternative information-theoretic measure that can quantify this difference?