Block Factorization of the Relative Entropy via Spatial Mixing

Convergence To Equilibrium ◽

Mixing Condition ◽

Entropy Functional ◽

Block Factorization

AbstractWe consider spin systems in the d-dimensional lattice $${\mathbb Z} ^d$$ Z d satisfying the so-called strong spatial mixing condition. We show that the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region $$V\subset {\mathbb Z} ^d$$ V ⊂ Z d in terms of a weighted sum of the entropies on blocks $$A\subset V$$ A ⊂ V when each A is given an arbitrary nonnegative weight $$\alpha _A$$ α A . These inequalities generalize the well known logarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type.

Logarithmic Sobolev inequalities in discrete product spaces

Combinatorics Probability Computing ◽

10.1017/s0963548319000099 ◽

2019 ◽

Vol 28 (06) ◽

pp. 919-935

Author(s):

Katalin Marton

Keyword(s):

Probability Measure ◽

Relative Entropy ◽

Sobolev Inequality ◽

Sobolev Inequalities ◽

Product Spaces ◽

Finite Set ◽

Local Specifications ◽

Image Position

AbstractThe aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*) $$D(p||q){\rm{\le}}C \cdot \sum\limits_{i = 1}^n {{\rm{\mathbb{E}}}_p D(p_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,...,{\rm{ }}Y_n )||q_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,{\rm{ }}...,{\rm{ }}Y_n )),} $$ where pi(· |y1, …, yi−1, yi+1, …, yn) and qi(· |x1, …, xi−1, xi+1, …, xn) denote the local specifications for p resp. q, that is, the conditional distributions of the ith coordinate, given the other coordinates. The constant C depends on (the local specifications of) q.The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy.In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.

Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

Annales Henri Poincaré ◽

10.1007/s00023-021-01088-3 ◽

2021 ◽

Author(s):

Ivan Bardet ◽

Ángela Capel ◽

Cambyse Rouzé

Keyword(s):

Relative Entropy ◽

Von Neumann Algebras ◽

Spin Systems ◽

Von Neumann ◽

Strong Subadditivity ◽

Finite Dimensional ◽

Lattice Spin ◽

Lattice Spin Systems ◽

Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

Reviews in Mathematical Physics ◽

10.1142/s0129055x94000407 ◽

1994 ◽

Vol 06 (05a) ◽

pp. 1147-1161 ◽

Cited By ~ 97

Author(s):

MARY BETH RUSKAI

Keyword(s):

Relative Entropy ◽

Discrete Case ◽

Sobolev Inequalities ◽

Completely Positive ◽

Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

Mixing heat-bath and Glauber dynamics: damage spreading in the Ising model

Journal of Physics A General Physics ◽

10.1088/0305-4470/22/22/006 ◽

1989 ◽

Vol 22 (22) ◽

pp. L1081-L1084 ◽

Cited By ~ 13

Author(s):

A M Mariz ◽

H J Herrmann

Keyword(s):

Ising Model ◽

Heat Bath ◽

Glauber Dynamics ◽

Damage Spreading

Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-2015-015-8 ◽

2015 ◽

Vol 67 (6) ◽

pp. 1384-1410 ◽

Cited By ~ 1

Author(s):

Piotr Graczyk ◽

Todd Kemp ◽

Jean-Jacques Loeb

Keyword(s):

Large Class ◽

Exponential Type ◽

Sobolev Inequality ◽

Density Theorem ◽

Sobolev Inequalities ◽

Subharmonic Functions ◽

Compactly Supported

AbstractWe prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.

REMARKS ON THE FREE RELATIVE ENTROPY AND THE FREE FISHER INFORMATION DISTANCE

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025710004048 ◽

2010 ◽

Vol 13 (02) ◽

pp. 243-255

Author(s):

HIROAKI YOSHIDA

Keyword(s):

Relative Entropy ◽

Fisher Information ◽

Sobolev Inequality ◽

Semicircle Law ◽

Information Distance ◽

Poisson Law ◽

Free Relative ◽

Free Fisher Information

In this paper, we shall introduce the free Fisher information distance which is inspired by the estimation-theoretic representation of the free relative entropy investigated by Verdú. We shall see the free analogue of the logarithmic Sobolev inequality with respect to a centered semicircle law and also the semicircular approximation of the free Poisson law.

Competing Glauber and Kawasaki Dynamics

International Journal of Modern Physics B ◽

10.1142/s0217979298001393 ◽

1998 ◽

Vol 12 (23) ◽

pp. 2385-2392 ◽

Cited By ~ 9

Author(s):

Simone Artz ◽

Steffen Trimper

Keyword(s):

Correlation Length ◽

Heat Bath ◽

Glauber Dynamics ◽

Long Range Order ◽

Strongly Correlated ◽

Kawasaki Dynamics ◽

Kinetic Ising Model ◽

Ginzburg Landau ◽

Static Interaction ◽

Quantum Formulation

Using a quantum formulation of the master equation we study a kinetic Ising model with competing stochastic processes: the Glauber dynamics with probability p and the Kawasaki dynamics with probability 1-p. Introducing explicitly the coupling to a heat bath and the mutual static interaction of the spins the model can be traced back exactly to a Ginzburg–Landau functional when the interaction is of long range order. The dependence of the correlation length on the temperature and on the probability p is calculated. In case that the spins are subject to flip processes the correlation length disappears for each finite temperature. In the exchange dominated case the system is strongly correlated for each temperature.