Quantum Talagrand-type Inequalities via Variance Decay

TL;DR

This paper develops a unified, dimension-free framework for quantum Talagrand inequalities on the quantum Boolean cube, connecting variance decay, hypercontractivity, and isoperimetric bounds.

Contribution

It introduces a variance-decay perspective that unifies first-order and high-order quantum Talagrand inequalities with explicit constants.

Findings

Derived quantum analogues of classical Talagrand inequalities.

Established high-order quantum Talagrand--KKL-type bounds.

Provided a broad, unified variance-decay framework for quantum inequalities.

Abstract

We establish dimension-free quantum Talagrand-type inequalities with explicit constants on the quantum Boolean cube, via a unified variance-decay perspective. For individual observables, short-time variance decay along the depolarizing semigroup, with rates estimated through hypercontractivity, naturally yields Talagrand-type bounds. Within this framework, we derive Talagrand-type energy--variance and high-order influence--variance inequalities. From the former, we obtain quantum analogues of Talagrand's isoperimetric inequality, the Eldan--Gross inequality, and the Cordero-Erausquin--Eskenazis inequality; from the latter, we derive high-order quantum Talagrand--KKL-type and partial isoperimetric bounds. Altogether, our work provides a variance-decay framework of broad applicability, unifying first-order and high-order Talagrand-type phenomena.