  We relate factorizable quantum channels on $M_n$, for $n \ge 2$, via their
Choi matrix, to certain correlation matrices, which, in turn, are shown to be
parametrized by traces on the unital free product $M_n * M_n$. Factorizable
maps that admit a finite dimensional ancilla are parametrized by finite
dimensional traces on $M_n * M_n$, and factorizable maps that approximately
factor through finite dimensional C*-algebras are parametrized by traces in the
closure of the finite dimensional ones. The latter set is shown to be equal to
the set of hyperlinear traces on $M_n * M_n$. We finally show that each
metrizable Choquet simplex is a face of the simplex of tracial states on $M_n *
M_n$.
