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# On linear-algebraic notions of expansion

Dec 2022

A fundamental fact about bounded-degree graph expanders is that three notionsof expansion -- vertex expansion, edge expansion, and spectral expansion -- areall equivalent. In this paper, we study to what extent such a statement is truefor linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namelydimension expansion (defined in analogy to graph vertex expansion) and quantumexpansion (defined in analogy to graph spectral expansion). Lubotzky andZelmanov proved that the latter implies the former. We prove that the converseis false: there are dimension expanders which are not quantum expanders. Moreover, this asymmetry is explained by the fact that there are two distinctlinear-algebraic analogues of graph edge expansion. The first of these isquantum edge expansion, which was introduced by Hastings, and which he provedto be equivalent to quantum expansion. We introduce a new notion, termeddimension edge expansion, which we prove is equivalent to dimension expansionand which is implied by quantum edge expansion. Thus, the separation above isimplied by a finer one: dimension edge expansion is strictly weaker thanquantum edge expansion. This new notion also leads to a new, more modular proofof the Lubotzky--Zelmanov result that quantum expanders are dimensionexpanders.

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