We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver [A von Neumann algebra approach to quantum metrics, Mem. Am. Math. Soc. 215(1010) (2012) 1–80]. 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 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 from [K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf and F. Verstraete, The ${\chi }^2$-divergence and mixing times of quantum Markov processes, J. Math. Phys. 51(12) (2010) 122201].

我们在 Kuperberg 和 Weaver [A von Neumann algebra approach to quantum metrics，Mem 。是。数学。社会。 215 (1010) (2012) 1–80]。我们表明，量子渐近维度在度量商和直和方面表现良好，并且在量子粗嵌入下得以保留。此外，我们证明了等粗包含扩展子序列的量子度量空间必须具有无穷大的渐近维数。这是通过证明扩展器的顶点等周不等式的量子版本来完成的，基于先前已知的来自 [K. Temme、MJ Kastoryano、MB Ruskai、MM Wolf 和 F. Verstraete，The${\chi }^{\mathrm{2个}}$-量子马尔可夫过程的发散和混合时间，J. Math。物理。 51 (12) (2010) 122201]。