2015/06/30 by Hao-Chung Cheng, Min-Hsiu Hsieh · 4 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Bounding overwatch #Equivalence (formal languages) #Gaussian #Hypercube #Markov chain #Mathematical Inequalities and Applications #Matrix (chemical analysis) #Quantum #Statistical Mechanics and Entropy #Unitary matrix #Unitary state #Wireless Communication Security Techniques #cs.IT #math-ph #math.IT #math.MP #math.PR #quant-ph
paper · pdf · doi:10.1063/1.5035381
published in Journal of Mathematical Physics 60(3) (American Institute of Physics)
openalex publication_date 2019/03/01 · openalex created_date 2019/03/22 · arxiv created 2019/05/02 · arxiv updated 2019/05/06 · openalex updated_date 2026/08/05
Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative Lp 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.