Quantum expanders and geometry of operator spaces

  • Gilles Pisier

    Texas A&M University, College Station, USA


We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the ``growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of MNM_N-spaces needed to represent (up to a constant C>1C>1) the MNM_N-version of the nn-dimensional operator Hilbert space OHnOH_n as a direct sum of copies of MNM_N. We show that, when CC is close to 1, this multiplicity grows as expβnN2\exp{\beta n N^2} for some constant β>0\beta>0. The main idea is to relate quantum expanders with "smooth" points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on nn-dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.

Cite this article

Gilles Pisier, Quantum expanders and geometry of operator spaces. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1183–1219

DOI 10.4171/JEMS/458