Convergence rate for spectral distribution of addition of random matrices

Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H=A+UBU⁎ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate 1N in the bulk of the spectrum.