  Sobolev-type inequalities have been extensively studied in the frameworks of
real-valued functions and non-commutative $\mathbb{L}_p$ spaces, and have
proven useful in bounding the time evolution of classical/quantum Markov
processes, among many other applications. In this paper, we consider yet
another fundamental setting - matrix-valued functions - and prove new
Sobolev-type inequalities for them. Our technical contributions are two-fold:
(i) we establish a series of matrix Poincar\'e inequalities for separably
convex functions and general functions with Gaussian unitary ensembles inputs;
and (ii) we derive $\Phi$-Sobolev inequalities for matrix-valued functions
defined on Boolean hypercubes and for those with Gaussian distributions. Our
results recover the corresponding classical inequalities (i.e.~real-valued
functions) when the matrix has one dimension. Finally, as an application of our
technical outcomes, we derive the upper bounds for a fundamental entropic
quantity - the Holevo quantity - in quantum information science since
classical-quantum channels are a special instance of matrix-valued functions.
This is obtained through the equivalence between the constants in the strong
data processing inequality and the $\Phi$-Sobolev inequality.
