Talagrand-type inequalities are fundamental in Boolean function analysis, providing sharp, dimension-free control of variance in terms of energy and influence, with applications to learning theory. In this talk, I will present a unified semigroup-based framework for proving such inequalities in the quantum Boolean setting. The framework is built on a simple two-step principle: short-time variance decay along the depolarizing semigroup, with rates estimated via hypercontractivity, yields Talagrand-type bounds with explicit, dimension-free constants. This approach recovers quantum analogues of several classical results, including the Eldan–Gross inequality and Talagrand's influence inequality.

I will also discuss ongoing work on high-order quantum extensions. Building on Besov-type variance functionals recently developed for classical high-order Talagrand inequalities, we are extending the variance-decay framework to arbitrary order. This will yield quantum analogues of high-order influence and isoperimetric inequalities, with potential applications to learning quantum Boolean functions with k-body interactions.

Based on arXiv:2601.01900 and arXiv:2606.14876.

报告人简介：李沛杰于2025年在香港大学获得数学博士学位，目前是香港大学数学系的博士后研究员。其主要研究方向包括概率论、信息论、布尔函数分析、泛函不等式。