Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

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 Poincare 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.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 Poincare 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 sin...

[1]  R. Bhatia.  Matrix Analysis , 1996 .

[2]  S. Beigi,et al.  Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm , 2015, 1509.02610.

[3]  Ronald de Wolf,et al.  A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[4]  Christopher King.  Hypercontractivity for Semigroups of Unital Qubit Channels , 2012 .

[5]  L. Gross.  Existence and uniqueness of physical ground states , 1972 .

[6]  Daniel Stilck França,et al.  Sandwiched Rényi Convergence for Quantum Evolutions , 2016, 1607.00041.

[7]  Leonard Gross,et al.  Chapter Two. Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys , 2006 .

[8]  Marco Tomamichel,et al.  Exponential Decay of Matrix $Φ$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles , 2015, ArXiv.

[9]  B. Zegarliński,et al.  Hypercontractivity in Noncommutative LpSpaces , 1999 .

[10]  Leonard Gross.  Logarithmic sobolev inequalities — A survey , 1978 .

[11]  L. Gross.  Logarithmic Sobolev inequalities and contractivity properties of semigroups , 1993 .

[12]  A. Bonami.  Étude des coefficients de Fourier des fonctions de $L^p(G)$ , 1970 .

[13]  P. Federbush.  Partially Alternate Derivation of a Result of Nelson , 1969 .

[14]  Igal Sason,et al.  Concentration of Measure Inequalities in Information Theory, Communications, and Coding , 2012, Found. Trends Commun. Inf. Theory.

[15]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[16]  R. Carbone,et al.  Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup , 2007 .

[17]  Nathan Srebro,et al.  Fast maximum margin matrix factorization for collaborative prediction , 2005, ICML.

[18]  S. Bobkov.  SOME EXTREMAL PROPERTIES OF THE BERNOULLI DISTRIBUTION , 1997 .

[19]  M. Ledoux.  On Talagrand's deviation inequalities for product measures , 1997 .

[20]  Massimiliano Pontil,et al.  Convex multi-task feature learning , 2008, Machine Learning.

[21]  Joel A. Tropp,et al.  An Introduction to Matrix Concentration Inequalities , 2015, Found. Trends Mach. Learn..

[22]  Joel A. Tropp,et al.  Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices , 2013, ArXiv.

[23]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[24]  Raffaella Carbone,et al.  Logarithmic Sobolev inequalities in non-commutative algebras , 2015 .

[25]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[26]  Maxim Raginsky,et al.  Logarithmic Sobolev inequalities and strong data processing theorems for discrete channels , 2013, 2013 IEEE International Symposium on Information Theory.

[27]  B. Zegarliński,et al.  Monotone norms and Finsler structures in noncommutative spaces , 2014 .

[28]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[29]  B. Zegarliński,et al.  Log-Sobolev inequalities for infinite one dimensional lattice systems , 1990 .

[30]  Ashley Montanaro,et al.  Quantum boolean functions , 2008, Chic. J. Theor. Comput. Sci..

[31]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[32]  J. Schur.  Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. , 1911 .

[33]  Ping-Cheng Yeh,et al.  The learnability of unknown quantum measurements , 2015, Quantum Inf. Comput..

[34]  Free Hypercontractivity , 1997 .

[35]  B. Zegarliński,et al.  Linear and Nonlinear Dissipative Dynamics , 2016 .

[36]  K. Temme,et al.  Quantum logarithmic Sobolev inequalities and rapid mixing , 2012, 1207.3261.

[37]  Raffaella Carbone.  OPTIMAL LOG-SOBOLEV INEQUALITY AND HYPERCONTRACTIVITY FOR POSITIVE SEMIGROUPS ON $M_2({\mathbb C})$ , 2004 .

[38]  Boguslaw Zegarlinski,et al.  HYPERCONTRACTIVITY VIA SPECTRAL THEORY , 2000 .

[39]  V. V. Peller,et al.  Hankel operators in the perturbation theory of unitary and self-adjoint operators , 1985 .

[40]  R. Latala,et al.  Between Sobolev and Poincaré , 2000, math/0003043.

[41]  L. Gross.  LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .

[42]  D. Petz,et al.  Families of completely positive maps associated with monotone metrics , 2012, 1212.1337.

[43]  Freeman J. Dyson,et al.  The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics , 1962 .

[44]  J. Tropp,et al.  Efron–Stein inequalities for random matrices , 2014, 1408.3470.

[45]  Alice Guionnet,et al.  Lectures on Logarithmic Sobolev Inequalities , 2003 .

[46]  T. Cubitt,et al.  Quantum reverse hypercontractivity , 2015, 1504.06143.

[47]  D. Bakry.  L'hypercontractivité et son utilisation en théorie des semigroupes , 1994 .

[48]  C. Tebaldi,et al.  Option pricing with Correlation Risk , 2007 .

[49]  Damir Filipović,et al.  Affine Processes on Positive Semidefinite Matrices , 2009, 0910.0137.

[50]  Kelly Bickel,et al.  Differentiating Matrix Functions , 2011, 1104.0336.

[51]  A. Montanaro.  Some applications of hypercontractive inequalities in quantum information theory , 2012, 1208.0161.

[52]  Pierre Bernard,et al.  Lectures on probability theory , 1994 .

[53]  Eric Carlen,et al.  Optimal hypercontractivity for fermi fields and related non-commutative integration inequalities , 1992, hep-th/9209053.

[54]  Maxim Raginsky,et al.  Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels , 2014, IEEE Transactions on Information Theory.

[55]  Daniel Stilck França,et al.  Relative Entropy Convergence for Depolarizing Channels , 2015, 1508.07021.

[56]  M. Yuan,et al.  Model selection and estimation in regression with grouped variables , 2006 .

[57]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[58]  Michael M. Wolf,et al.  Entropy Production of Doubly Stochastic Quantum Channels , 2015 .

[59]  F. Verstraete,et al.  The χ2-divergence and mixing times of quantum Markov processes , 2010, 1005.2358.

[60]  K. Atkinson,et al.  Theoretical Numerical Analysis: A Functional Analysis Framework , 2001 .

[61]  B. Simon,et al.  Hypercontractive semigroups and two dimensional self-coupled Bose fields , 1972 .

[62]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[63]  Douglas Farenick,et al.  Jensen's inequality relative to matrix-valued measures , 2007 .

[64]  T. Tao.  Topics in Random Matrix Theory , 2012 .

[65]  Min-Hsiu Hsieh,et al.  Characterizations of matrix and operator-valued Φ-entropies, and operator Efron–Stein inequalities , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.