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.