Fisher Information and Logarithmic Sobolev Inequality for Matrix Valued Functions

TL;DR

This paper establishes a concentration inequality for matrix-valued functions on manifolds using noncommutative geometry and demonstrates the density of generators satisfying a modified logarithmic Sobolev inequality in finite-dimensional matrix algebras, with implications for quantum information.

Contribution

It extends Talagrand's concentration inequality to matrix-valued functions on manifolds and shows the density of certain generators satisfying a modified logarithmic Sobolev inequality.

Findings

Proves a matrix-valued concentration inequality on Riemannian manifolds.

Shows density of generators satisfying the inequality in finite-dimensional matrix algebras.

Links noncommutative geometry tools to quantum information theory.

Abstract

We prove a version of Talagrand's concentration inequality for subordinated sub-Laplacian on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.