Quantum Talagrand-type Inequalities via Variance Decay
Fan Chang, Peijie Li
TL;DR
This work establishes a dimension-free Talagrand-type variance inequality on the quantum Boolean cube $M_2(\mathbb{C})^{\otimes n}$ by introducing an $\alpha$-interpolated local gradient $|\nabla_j^\alpha A|^2=(1-\alpha)\mathrm{Var}_j(A)+\alpha|d_jA|^2$ whose squared mass $|\nabla^\alpha A|^2$ remains invariant in $L^2$-mass. The main result provides a lower bound $\|A\|_{\infty}^{2-p}\,\bigl\||\nabla^{\alpha}A|\bigr\|_p^{p} \gtrsim \mathrm{Var}(A)\cdot \max\{1,\mathcal{R}(A,q)^{p/2}\}$ with a logarithmic ratio $\mathcal{R}(A,q)$ that captures how small either the centered function $A-\tau(A)$ or the gradient vector is in $L^q$ relative to the variance. Consequences include a quantum Eldan–Gross inequality, a quantum Cordero–Erausquin–Eskenazis $L^p$-$L^q$ inequality, and quantum isoperimetric bounds, all derived via a semigroup approach that leverages improved Lipschitz smoothing estimates from noncommutative Khintchine theory. The paper further develops a high-order theory with a local variance functional $V_J(A)$ and proves local Talagrand-type inequalities for high-order influences, yielding $L^p$-$L^q$ influence inequalities and partial isoperimetric bounds for $k$-wise influences. Overall, the results extend classical influence/isoperimetric phenomena to the noncommutative quantum cube with dimension-free, semigroup-based proofs and sharpened smoothing estimates, highlighting the role of local structure and high-order interactions in quantum Boolean analysis.
Abstract
We establish dimension-free Talagrand-type variance inequalities on the quantum Boolean cube $M_{2}(\mathbb C)^{\otimes n}$. Motivated by the splitting of the local carré du champ into a conditional-variance term and a pointwise-derivative term, we introduce an $α$-interpolated local gradient $|\nabla_j^αA|$ that bridges $\mathrm{Var}_j(A)$ and $|d_jA|^{2}$. For $p\in[1,2],q\in[1,2)$ and $α\in[0,1]$, we prove a Talagrand-type inequality of the form $$\|A\|_\infty^{2-p}\,\bigl\||\nabla^αA|\bigr\|_{p}^{p}\ \gtrsim\mathrm{Var}(A)\cdot \max\left\{1, \mathcal{R}(A,q)^{p/2}\right\},$$ where $\mathcal{R}(A,q)$ is a logarithmic ratio quantifying how small either $A-τ(A)$ or the gradient vector $(d_jA)_j$ is in $L^{q}$ compared to $\mathrm{Var}(A)^{1/2}$. As consequences we derive a quantum Eldan--Gross inequality in terms of the squared $\ell_2$-mass of geometric influences, a quantum Cordero-Erausquin--Eskenazis $L^{p}$-$L^{q}$ inequality, and Talagrand-type $L^{p}$-isoperimetric bounds. We further develop a high-order theory by introducing the local variance functional $$V_J(A)=\int_0^\infty 2\mathrm{Inf}^{2}_{J}(P_tA) dt.$$ For $|J|=k$ we prove a local high-order Talagrand inequality relating $\mathrm{Inf}^{p}_{J}[A]$ to $V_J(A)$, with a Talagrand-type logarithmic term when $\mathrm{Inf}^{q}_{J}[A]$ is small. This yields $L^{p}$-$L^{q}$ influence inequalities and partial isoperimetric bounds for high-order influences. Our proofs are purely semigroup-based, relying on an improved Lipschitz smoothing estimate for $|\nabla^αP_tA|$ obtained from a sharp noncommutative Khintchine inequality and hypercontractivity.