Title: Algebraic formulations of Zauner's conjecture
| Speaker: | Jon Yard |
| Affliliation: | University of Waterloo |
| Zoom: | Contact Emma Watson |
Abstract:
Tight complex projective 2-designs are simultaneously maximal sets of equiangular lines and minimal complex projective 2-designs. In quantum information theory, they define optimal measurements known as SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures). They are conjectured by Zauner to exist in every dimension, even as specific group orbits. Yet, they have only so far been proven to exist in a finite-but-growing list of dimensions via exact, explicit constructions over increasingly high-degree number fields, since identified as specific class fields of real quadratic number fields. In this talk, I will reformulate Zauner's conjecture on their existence in terms of ideals of invariant quadratic harmonic functions on complex projective space, showing how it sheds light on algebraic properties of the solutions. Implications for Hilbert's 12th problem on the explicit generation of such number fields will also be discussed.