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

Operator-Valued Matrices with Free or Exchangeable Entries