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You are here: Home en Opencourse SGU Mathematics Kickoff Meeting video Quantum expanders, random matrices and geometry of operator spaces

Quantum expanders, random matrices and geometry of operator spaces



Abstract
Using random unitary matrices, 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 of “geometric” nature, for the operator space analogue of Euclidean geometry. This allows us to provide sharp estimates for the growth of the multiplicity of MN-spaces needed to represent (up to a constant C > 1) the MN -version of the n-dimensional operator Hilbert space OHn as a direct sum of copies of MN. We show that, when C is close to 1, this multiplicity grows as expβnN2 for some constant β > 0. The main idea is to identify quantum expanders with “smooth” points on the matricial analogue of the unit sphere, and to show that there are plenty of “uniformly smooth” points (more precisely as many as allowed by a soft metric entropy dimensional restriction). This generalizes to operator spaces a classical geometric result on n-dimensional Hilbert space (corresponding to N = 1). Our work strongly suggests to further study a certain class of operator spaces that we call matricially subGaussian.
(About the formula, please refer to https://www.math.kyoto-u.ac.jp/sites/default/files/public_files/SGUKick_0.pdf.)