Quantum KKL-type Inequalities Revisited

TL;DR

This paper extends classical inequalities to the quantum setting using a novel random restriction method, establishing quantum analogs of key inequalities and providing alternative insights into the quantum KKL conjecture.

Contribution

It introduces a quantum version of the random restriction method and proves several quantum inequalities, advancing understanding of quantum isoperimetric and KKL-type problems.

Findings

Established quantum Eldan-Gross inequality

Proved quantum Talagrand isoperimetric inequality

Reproduced recent results on quantum KKL conjecture

Abstract

In the present paper, we develop the random restriction method in the quantum framework. By applying this method, we establish the quantum Eldan-Gross inequality, the quantum Talagrand isoperimetric inequality, and related quantum KKL-type inequalities. Our results recover some recent results of Rouz\'e et al. \cite{RWZ2024} and Jiao et al. \cite{JLZ2025}, which can be viewed as alternative answers to the quantum KKL conjecture proposed by Motanaro and Osborne in \cite{MO2010}.