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We provide bounds of Berry-Esseen type for fundamental limit theorems inoperator-valued free probability theory such as the operator-valued freeCentral Limit Theorem and the asymptotic behaviour of distributions ofoperator-valued matrices. Our estimates are on the level of operator-valuedCauchy transforms and the L\'evy distance. We address the single-variable aswell as the multivariate setting for which we consider linear matrix pencilsand noncommutative polynomials as test functions. The estimates are in terms ofoperator-valued moments and yield the first quantitative bounds on the L\'evydistance for the operator-valued free Central Limit Theorem. Our results alsoyield quantitative estimates on joint noncommutative distributions ofoperator-valued matrices having a general covariance profile. In thescalar-valued multivariate case, these estimates could be passed to explicitbounds on the order of convergence under the Kolmogorov distance.

Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices