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We consider random matrices of the form   $H_N=A_N+U_N B_N U^*_N$  , where   $A_N$   and   $B_N$   are two   $N$   by   $N$   deterministic Hermitian matrices and   $U_N$   is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of   $H_N$   on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of   $H_N$   and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.

Central Limit Theorem for Mesoscopic Eigenvalue Statistics of the Free Sum of Matrices