Free Malliavin-Stein-Dirichlet method: multidimensional semicircular approximations and chaos of a quantum Markov operator

We combine the notion of free Stein kernel and the free Malliavin calculus to
provide quantitative bounds under the free (quadratic) Wasserstein distance in
the multivariate semicircular approximations for self-adjoint vector-valued
multiple Wigner integrals. On the way, we deduce an HSI inequality for a
modified non-microstates free entropy with respect to the potential associated
with these semicircular families in the case of non-degeneracy of the
covariance matrix. The strategy of the proofs is based on functional
inequalities involving the free Stein discrepancy. We obtain a bound which
depends on the second and fourth free cumulant of each component. We then apply
these results to some examples such as the convergence of marginals in the free
functional Breuer-Major CLT for the non commutative fractional Brownian motion,
and we provide a bound for the free Stein discrepancy with respect to
semicircular potentials for $q$-semicirculars operators}. Lastly, we develop an
abstract setting on where it is possible to construct a free Stein Kernel with
respect to the semicircular potential: the quantum chaos associated to a
quantum Markov semigroup whose $L^2$ generator $\Delta$ can be written as the
square of a real closable derivation $\delta$ valued into the square integrable
bi-processes or into a direct sum of the coarse correspondence.