  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.
