Title: The number theory of equiangular lines
| Speaker: | Jon Yard |
| Affiliation: | University of Waterloo |
| Room: | MC 5501 |
Abstract:
It is easy to prove that there can exist at most d2 equiangular complex lines in Cd. Configurations saturating this bound are known by other names: maximal equiangular tight frames, minimal complex projective 2-designs and symmetric informationally complete positive operator-valued measures (SIC-POVMs). Zauner conjectured that sets of d2 equiangular lines exist in every Cd. So far, this has been proved for a growing (but still finite) list of dimensions, where explicit constructions have been found as orbits of finite Heisenberg groups via extensive computations over high-degree number fields.
In this talk, I will start by giving some background, emphasizing low-dimensional examples. Then I will discuss the evidence (found with Appleby, Flammia and McConnell) that the known explicit constructions possess combined Galois and unitary symmetries that can be explained by the mathematics of class field theory, providing further evidence in support of Zauner's conjecture. The relevant number fields are certain ray class fields of real quadratic number fields, illustrating that the existence of equiangular Heisenberg orbits is intimately related to Hilbert's 12th problem of finding explicit generators for abelian extensions of real quadratics.