Asymptotic dimension and coarse embeddings in the quantum setting

TL;DR

This paper extends classical concepts of asymptotic dimension and coarse embeddings to quantum metric spaces, revealing their behavior and limitations in the quantum setting, especially regarding quantum expanders.

Contribution

It introduces quantum asymptotic dimension, analyzes its properties, and establishes a quantum vertex-isoperimetric inequality related to quantum expanders.

Findings

Quantum asymptotic dimension is preserved under quantum coarse embeddings.

Quantum asymptotic dimension is infinite for spaces containing quantum expanders.

A quantum vertex-isoperimetric inequality is proven for quantum expanders.

Abstract

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete.