  We study matrices whose entries are free or exchangeable noncommutative
elements in some tracial $W^*$-probability space. More precisely, we consider
operator-valued Wigner and Wishart matrices and prove quantitative convergence
to operator-valued semicircular elements over some subalgebra in terms of
Cauchy transforms. As direct applications, we obtain explicit rates of
convergence for a large class of random block matrices with independent or
correlated blocks. Our approach relies on a noncommutative extension of the
Lindeberg method and operator-valued Gaussian interpolation techniques.
