Quantum Expanders and Geometry of Operator Spaces

TL;DR

This paper constructs large families of quantum expanders and explores their implications for the growth of operator spaces, providing sharp estimates and connecting quantum expanders with geometric properties of operator spaces.

Contribution

It introduces well-separated quantum expanders with maximal size and links them to the geometry of operator spaces, extending classical geometric results to the quantum setting.

Findings

Existence of large families of quantum expanders with maximal cardinality

Asymptotically sharp estimates for the growth of operator space multiplicities

Connection between quantum expanders and geometric properties of operator spaces

Abstract

We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the "growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_N$-spaces needed to represent (up to a constant $C>1$) the $M_N$-version of the $n$-dimensional operator Hilbert space $OH_n$ as a direct sum of copies of $M_N$. We show that, when $C$ is close to 1, this multiplicity grows as $\exp{\beta n N^2}$ for some constant $\beta>0$. The main idea is to relate quantum expanders with "smooth" points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on $n$-dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.