  Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with compact
support, and let $\mu \boxplus \nu$ denote their additive free convolution. We
show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of
$\mu$ and $\nu$, we have \begin{equation*} \int_{-\infty}^\infty \log(z - x)
\mu \boxplus \nu (\mathrm{d}x) = \sup_{\Pi} \left\{ \mathbf{E}_\Pi[\log(z -
(X+Y)] - H(\Pi|\mu \otimes \nu) \right\}, \end{equation*} where the supremum is
taken over all couplings $\Pi$ of the probability measures $\mu$ and $\nu$, and
$H(\Pi|\mu \otimes \nu)$ denotes the relative entropy of a coupling $\Pi$
against product measure. We prove similar formulas for the multiplicative free
convolution $\mu \boxtimes \nu$ and the free compression $[\mu]_\tau$ of
probability measures, as well as for multivariate free operations. Thus the
integrals of a log-potential against the fundamental measure operations of free
probability may be formulated in terms of entropic optimal transport problems.
The optimal couplings in these variational descriptions of the free probability
operations can be computed explicitly, and from these we can then deduce the
standard $R$- and $S$-transform descriptions of additive and multiplicative
free convolution. We use our optimal transport formulations to derive new
inequalities relating free and classical operations on probability measures,
such as the inequality \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu
\boxplus \nu (\mathrm{d}x) \geq \int_{-\infty}^{\infty} \log(z-x) \mu \ast \nu(
\mathrm{d}x) \end{equation*} relating free and classical convolution. Our
approach is based on applying a large deviation principle on the symmetric
group to the quadrature formulas of Marcus, Spielman and Srivastava.
