Quantum Expanders and Geometry of Operator Spaces II

TL;DR

This paper presents a simplified inequality related to quantum expanders, showing that moments of certain operator norms are dominated by Gaussian sums, which simplifies analysis and broadens applicability.

Contribution

It introduces a less sharp but more general inequality for moments of operator norms involving quantum expanders and Gaussian sums.

Findings

Moments of the operator norm are dominated by Gaussian sums.

The inequality applies to more general (including matricial) coefficients.

Gaussian sums are easier to analyze due to independence and Gaussian properties.

Abstract

In this appendix to our paper with the same title posted on arxiv we give a quick proof of an inequality that can be substituted to Hastings's result, quoted as Lemma 1.9 in our previous paper. Our inequality is less sharp but also appears to apply with more general (and even matricial) coefficients. It shows that up to a universal constant all moments of the norm of a linear combination of the form $$S=\sum\nolimits_j a_j U_j \otimes \bar U_j (1-P)$$ are dominated by those of the corresponding Gaussian sum $$S'=\sum\nolimits_j a_j Y_j \otimes \bar Y'_j .$$ The advantage is that $S'$ is now simply separately a Gaussian random variable with respect to the independent Gaussian random matrices $(Y_j)$ and $(Y'_j)$, and hence is much easier to majorize. Note we plan to incorporate this appendix into our future publication.