Algebraic Graph Theory Seminar - Jon Yard

Title: Equiangular complex projective 2-designs

Abstract: Tight 2-designs in complex projective space correspond to maximal sets of complex equiangular lines.  They are also known as equiangular tight frames in signal processing and as SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) in quantum information theory.  While they are believed to exist in every finite-dimensional complex Hilbert space as orbits of finite Heisenberg groups, their existence has so far only been mathematically proven for finitely many dimensions.  The proofs in these cases are mainly computational and require computer-assisted calculations in number fields of degree growing roughly quadratically with the dimension.  In this talk, I will illustrate how the structure of these number fields can be explained using class field theory.  The resulting uniformity of these fields constitutes overwhelming evidence in favor of their existence in every dimension and uncovers a surprising connection to Hilbert's still-unsolved 12th problem of giving explicit generators for class fields of real quadratic fields. 

This talk is partially based on joint work (arxiv:1604.06098) with Appleby, Flammia and McConnell.