scholarly journals On Poincaré and Logarithmic Sobolev Inequalities for a Class of Singular Gibbs Measures

2020 ◽  
pp. 219-246
Author(s):  
Djalil Chafaï ◽  
Joseph Lehec
Keyword(s):  
Gibbs Measures ◽  
Sobolev Inequalities ◽  
Logarithmic Sobolev Inequalities

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

1994 ◽  
Vol 06 (05a) ◽  
pp. 1147-1161 ◽  
Author(s):  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Discrete Case ◽  
Sobolev Inequalities ◽  
Completely Positive ◽  
Logarithmic Sobolev Inequalities ◽  
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.

scholarly journals Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

2019 ◽  
Vol 178 (2) ◽  
pp. 319-378 ◽  
Author(s):  
Eric A. Carlen ◽  
Jan Maas
Keyword(s):  
Spectral Gap ◽  
Optimal Transport ◽  
Quantum Systems ◽  
Sobolev Inequalities ◽  
Unified Framework ◽  
Logarithmic Sobolev Inequalities ◽  
Curvature Bounds ◽  
Finite Dimensional ◽  
Dissipative Quantum Systems ◽  
Entropy Inequalities

AbstractWe study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $$C^*$$ C ∗ -algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

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