If $X$ and $Y$ are random variables, then a **maximal coupling** of $X$ and $Y$ is a coupling $\left(X', Y'\right)$ such that $\mathbf{P}\left(X'=Y'\right)$ is maximal (that is, the probability that the coupled variables coincide is optimal). By $``$coupling$"$, I'm referring to a random vector $\left(X',Y'\right)$ such that $X'$ has the same distribution as $X$, and $Y'$ has the same distribution as $Y$.



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> Are there any benefits of using a maximal coupling to describe distributional results regarding the marginals $X$ and $Y$? E.g., are the measures of dependence between $X$ and $Y$, such as covariance, mutual information, joint entropy, Kullback-Leibler distance, or $$\lvert\mathbf{P}\left(X'\in A\text{ and }Y'\in B\right)-\mathbf{P}\left(X'\in A\right)\mathbf{P}\left(Y'\in B\right)\rvert$$ going to be extreme in the maximal coupling?
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Measures of dependence in a maximal coupling
$\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$Let $\mu$ and $\nu$ stand for the distributions of real-valued random variables (r.v.'s) $X$ and $Y$, respectively. Let $\Pi(\mu,\nu)$ denote the set of all probability distributions over $\R^2$ with marginals $\mu$ and $\nu$.


By Strassen's [Theorem 11](https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-36/issue-2/The-Existence-of-Probability-Measures-with-Given-Marginals/10.1214/aoms/1177700153.full), $\pi\in\Pi(\mu,\nu)$ is the distribution of a maximal coupling iff
\begin{equation}
 \pi(D)=1-\|\mu-\nu\|,
\end{equation}
where $D:=\{(x,x)\colon x\in\R\}$ and $\|\mu-\nu\|$ is the total variation distance between $\mu$ and $\nu$.


It should be clear now that a maximal coupling is not unique. Indeed, let $(P,N)$ be the Hahn decomposition of the signed measure $\mu-\nu$ and then let $(\mu-\nu)\_+$ and $(\mu-\nu)\_-=(\nu-\mu)\_+$ be the Jordan positive and negative parts of $\mu-\nu$, so that $(\mu-\nu)\_+(A)=(\mu-\nu)(A\cap P)$ and $(\mu-\nu)\_-(A)=-(\mu-\nu)(A\cap N)$ for Borel $A\subseteq\R$. Let $\mu\wedge\nu:=\mu-(\mu-\nu)\_+=\nu-(\nu-\mu)\_+$. Let $\la$ be any measure on $\R^2$ with marginals $(\mu-\nu)\_+$ and $(\nu-\mu)\_+$. For all Borel $B\subseteq\R^2$, let
\begin{equation}
 \pi(B)=(\mu\wedge\nu)(\text{pr}(B\cap D))+\la(B), 
\end{equation}
where $\text{pr}(x,y):=x$ for $(x,y)\in\R^2$.
Then $\pi\in\Pi(\mu,\nu)$ and $\pi$ is the distribution of a maximal coupling. In particular, if $\mu$ and $\nu$ are mutually singular, then any $\pi\in\Pi(\mu,\nu)$ whatsoever is the distribution of a maximal coupling.


It should also be clear now that a maximal coupling is just a thing in itself, unlike the things you mentioned: the covariance, the mutual information, the joint entropy, the Kullback--Leibler distance.


The Kullback--Leibler distance actually depends only on the marginals $\mu$ and $\nu$; any choice of
$\pi\in\Pi(\mu,\nu)$ plays no role.


The covariance depends on Euclidean distances, which are of no consequence for a maximal coupling -- which latter depends on the Hamming distance $\rho(x,y):=1(x\ne y)$. The mutual information and the joint entropy also depend on the Hamming distance, and yet in general they attain their extremes not at a maximal coupling.


E.g., suppose that $\mu(\{x\})=P(X=x)=\dfrac x{m(2m+1)}\,1(x\in[2m])$ and $\nu(\{y\})=P(Y=y)=\dfrac {2m+1-y}{m(2m+1)}\,1(y\in[2m])$, where $m$ is a positive integer and $[n]:=\{1,\dots,n\}$ -- so that $(\mu\wedge\nu)(\{x\})=\dfrac {\min(x,2m+1-x)}{m(2m+1)}\,1(x\in[2m])$. So, for a maximal coupling $(X',Y')$ we have $$P(X'=Y')=(\mu\wedge\nu)(\R)=\dfrac{m+1}{2m+1}>\dfrac12.$$
On the hand, here a coupling $(X'',Y'')$ that maximizes the covariance and the mutual information (and minimizes the joint entropy) is given by the condition $Y''=2m+1-X''$ almost surely, so that $$P(X''=Y'')=0.\quad\Box$$
We see that the extremizers of the covariance, the mutual information, and the joint entropy are very different from maximal couplings.


