On linear-algebraic notions of expansion

TL;DR

This paper explores linear-algebraic analogues of graph expansion, revealing asymmetries and introducing a new notion called dimension edge expansion, which clarifies the relationship between different types of linear-algebraic expansion.

Contribution

It introduces dimension edge expansion, proves its equivalence to dimension expansion, and clarifies the separation between dimension and quantum expanders.

Findings

Quantum expansion implies dimension expansion.

Dimension expanders do not necessarily imply quantum expanders.

Dimension edge expansion is weaker than quantum edge expansion.

Abstract

A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: there are dimension expanders which are not quantum expanders. Moreover, this asymmetry is explained by the fact that there are two distinct linear-algebraic analogues of graph edge expansion. The first of these is quantum edge expansion, which was introduced by Hastings, and which he proved to be equivalent to quantum expansion. We introduce a new notion, termed dimension edge expansion, which we prove is equivalent to dimension expansion and which is implied by quantum edge expansion. Thus, the separation above is implied by a finer one: dimension edge expansion is strictly weaker than quantum edge expansion. This new notion also leads to a new, more modular proof of the Lubotzky--Zelmanov result that quantum expanders are dimension expanders.