Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail

Question

Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of generality in treating $X$ as a single variable $U$ that takes on $2^n$ states with the same probabilities (see below for a more rigourous interpretation). For example, $Y\perp Z|X \iff Y\perp Z|U$, and quantities such as entropy and mutual information remain unchanged.

My question: Are there any examples where this replacement "fails"? That is, some property that holds for $(X,Y,Z)$ but doesn't hold mutatis mutandis for $(U,Y,Z)$?

What I mean by "treating $X$ as a single variable $U$": More formally, let $\sigma:\{1,\ldots, 2^n\}\to \{0,1\}^n$ be a bijection and enumerate the $2^n$ possible states of $X$ by $\sigma$. We can define $U$ to be a random variable on $2^n$ states such that $$ P(U=k) = P(X=\sigma(k)). $$

Answer 1

One property distinguishing $(X,Y,Z)$ from $(U,Y,Z)$ is that the co-domain of the mapping $(X,Y,Z)$ is $C_1:=\{0,1\}^n\times\{0,1\}\times\{0,1\}$, whereas the co-domain of the mapping $(U,Y,Z)$ is $C_2:=\{1,\dots,2^n\}\times\{0,1\}\times\{0,1\}\ne C_1$.