Geometric Approach Towards Complete Logarithmic Sobolev Inequalities

Abstract

In this paper, we use the Carnot-Caratheodory distance from sub-Riemanian geometry to prove entropy decay estimates for all finite dimensional symmetric quantum Markov semigroups. This estimate is independent of the environment size and hence stable under tensorization. Our approach relies on the transference principle, the existence of $t$-designs, and the sub-Riemannian diameter of compact Lie groups and implies estimates for the spectral gap.

12 pages. A preliminary version. Comments are welcome!

References in corpus (3)

• Estimating the decoherence time using non-commutative Functional Inequalities
• Complete Sobolev Type Inequalities
• Graph Hörmander Systems