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arXiv · 2601.01900

Quantum Talagrand-type Inequalities via Variance Decay

Fan Chang·Peijie Li
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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.

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Fan Chang, Peijie Li. 2026-01-05. Quantum Talagrand-type Inequalities via Variance Decay. https://arxiv.org/abs/2601.01900

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