Quantum Expanders and Geometry of Operator Spaces II

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 $$S={\sum }_j{a}_j{U}_j\otimes {\bar{U}}_j(1-P)$$ are dominated by those of the corresponding Gaussian sum $$S^{\prime }={\sum }_j{a}_j{Y}_j\otimes {\bar{Y}}_j^{\prime }.$$ The advantage is that $S^{\prime }$ is now simply separately a Gaussian random variable with respect to the independent Gaussian random matrices $(Y_j)$ and $(Y_j^{\prime })$, and hence is much easier to majorize. Note we plan to incorporate this appendix into our future publication.

References in corpus (2)

• Random Unitaries Give Quantum Expanders
• Quantum expanders and the quantum entropy difference problem

Cited by in corpus (1)

• Examples of non exact 1-subexponential $C^{\ast }$-algebras