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Sobolev-type inequalities have been extensively studied in the frameworks ofreal-valued functions and non-commutative $\mathbb{L}_p$ spaces, and haveproven useful in bounding the time evolution of classical/quantum Markovprocesses, among many other applications. In this paper, we consider yetanother fundamental setting - matrix-valued functions - and prove newSobolev-type inequalities for them. Our technical contributions are two-fold:(i) we establish a series of matrix Poincar\'e inequalities for separablyconvex functions and general functions with Gaussian unitary ensembles inputs;and (ii) we derive $\Phi$-Sobolev inequalities for matrix-valued functionsdefined on Boolean hypercubes and for those with Gaussian distributions. Ourresults recover the corresponding classical inequalities (i.e.~real-valuedfunctions) when the matrix has one dimension. Finally, as an application of ourtechnical outcomes, we derive the upper bounds for a fundamental entropicquantity - the Holevo quantity - in quantum information science sinceclassical-quantum channels are a special instance of matrix-valued functions.This is obtained through the equivalence between the constants in the strongdata processing inequality and the $\Phi$-Sobolev inequality.

Matrix Poincar\'e, \Phi-Sobolev inequalities, and quantum ensembles