We study 2-cabled analogs of Voiculescu's trace and free Gibbs states on
Jones planar algebras. These states are traces on a tower of graded algebras
associated to a Jones planar algebra. Among our results is that, with a
suitable definition, finiteness of free Fisher information for planar algebra
traces implies that the associated tower of von Neumann algebras consists of
factors, and that the standard invariant of the associated inclusion is exactly
the original planar algebra. We also give conditions that imply that the
associated von Neumann algebras are non-$\Gamma$ non-$L^2$ rigid factors.