Motivated by quantum information theory, we study log-Sobolev inequalities for matrix valued functions on finite graphs and compact Lie groups. Using tools from noncommutative geometry we find a surprising link between graph Laplacians and sublaplacians on the orthogonal group. Simple combinatorial tools, namely the existence of a spanning tree, then allows us to find concrete lower bounds for spectral gaps. Joint work with H. Li and N. LaRacuente.