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If $V$ is sufficiently regular real-valued convex function of $m$ non-commutative self-adjoint variables, then there is free Gibbs law $\lambda_V$ for a non-commutative $m$-tuple $X = (X_1,\dots,X_m)$, which describes the large-$N$ behavior of tuples $X^{(N)} = (X_1^{(N)}, \dots, X_m^{(N)})$ of random matrices chosen according to the probability density $(1/Z^{(N)}) e^{-N^2 V(x)}\,dx$. We show that for a function $f$ that is well approximated by trace polynomials, the classical conditional expectation of $f(X^{(N)})$ given $X_1^{(N)}$, ..., $X_k^{(N)}$ for $k < m$ converges to the $W^*$-algebraic conditional expectation of $f(X)$ given $X_1$, ..., $X_k$. We also construct an isomorphism from $W^*(X_1,\dots,X_m)$ to the algebra $W^*(S_1,\dots,S_m)$ generated by a free semicircular family (i.e.\ a free group factor), which arises as the large $N$ limit of transport maps for probability measures on $M_N(\mathbb{C})_{sa}^m$. This transport can be made ``upper-triangular'' in the sense that $W^*(X_k,\dots,X_m)$ is mapped to $W^*(S_k,\dots,S_m)$ for each $k = 1, \dots, m$. At the same time, this transport map witnesses the Talagrand entropy-cost inequality for $\mu$ relative to the law of a semicircular family.

Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability