Geometric influences on quantum Boolean cubes

TL;DR

This paper explores the geometric aspects of quantum Boolean cubes, establishing bounds, inequalities, and relations involving $L_1$-influence, noise stability, and quantum extensions of classical theorems using advanced mathematical techniques.

Contribution

It introduces a dimension-free bound for $L_1$-influence, a quantum Talagrand inequality, and a quantitative relation between noise stability and $L^1$-influence in quantum Boolean cubes.

Findings

Dimension-free $L_1$-influence bound established.

Quantum Talagrand inequality derived.

Relation between noise stability and $L^1$-influence proved.

Abstract

In this work, we study three problems related to the $L_1$-influence on quantum Boolean cubes. In the first place, we obtain a dimension free bound for $L_1$-influence, which implies the quantum $L^1$-KKL Theorem result obtained by Rouze, Wirth and Zhang. Beyond that, we also obtain a high order quantum Talagrand inequality and quantum $L^1$-KKL theorem. Lastly, we prove a quantitative relation between the noise stability and $L^1$-influence. To this end, our technique involves the random restrictions method as well as semigroup theory.