  We propose a boundary regularity condition for the $M_n(\mathbb{C})$-valued
subordination functions in free probability to prove the local limit theorem
and delocalization of eigenvectors for 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.
