Complete Sobolev Type Inequalities

TL;DR

This paper extends Sobolev inequalities to noncommutative quantum settings, generalizing classical inequalities using monotone metrics and exploring applications in quantum models.

Contribution

It introduces a framework for Sobolev inequalities in noncommutative spaces, broadening their applicability to quantum information theory.

Findings

Generalized monotone metrics in quantum state spaces

Established Sobolev inequalities in noncommutative contexts

Applied results to quantum models like random transpositions

Abstract

We establish Sobolev type inequalities in the noncommutative settings by generalizing monotone metrics in the space of quantum states, such as matrix-valued Beckner inequalities. We also discuss examples such as random transpositions and Bernoulli-Laplace models.