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We propose a boundary regularity condition for the   $M_n({\mathbb{C}})$  -valued subordination functions in free probability to prove a local limit theorem and delocalization of eigenvectors for self-adjoint polynomials in two random matrices. We prove this through estimating the pair of   $M_n({\mathbb{C}})$  -valued approximate subordination functions for the sum of two   $M_n({\mathbb{C}})$  -valued random matrices   $\gamma _1\otimes C_N+\gamma _2\otimes U_N^*D_NU_N$  , where   $C_N$  ,   $D_N$   are deterministic diagonal matrices, and   $U_N$   is Haar unitary.

A Local Limit Theorem and Delocalization of Eigenvectors for Polynomials in Two Matrices