  We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform
modified log-Sobolev inequality for all its matrix-valued functions,
independent of the matrix dimension. This is the first example of sub-Laplacian
that a matrix-valued modified log-Sobolev inequality has been obtained. We also
show that on Lie groups, the heat kernel measure $p_t$ at time $t$ admits
matrix-valued modified log-Sobolev constants of order $O(t^{-1})$.
