Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative 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 Poincaré inequalities for separably convex functions and general functions with Gaussian unitary ensembles inputs; and (ii) we derive Φ-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 Φ-Sobolev inequality.
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The Gaussian unitary ensembles are a family of random Hermitian matrices whose upper-triangular entries are independently and identically distributed (i.i.d.) complex standard Gaussian random variables, while the diagonal entries are i.i.d. real standard Gaussian random variables; see, e.g., Ref. 66, Sec. 2.6.
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