Title: Complex Equiangular Lines: The SIC-POVM problem
Abstract: SIC-POVMs are generalised quantum measurements which are of particular interest in the context of quantum state tomography and quantum key distribution. Alternatively, they can be described by d^2 unit vectors in the d-dimensional complex vectors space such that the inner product between any pair of vectors has constant modulus. They also form maximal equiangular tight frames.
In his doctoral thesis from 1999, Zauner conjectured that SIC-POVMs existed for all finite dimensions. Moreover, he conjecture that they can be obtained as the projective orbit of a fiducial vector with respect to the Weyl-Heisenberg group. Despite a lot of effort, numerical or exact fiducial vectors are only known for a finite, albeit growing list of dimensions. Currently, numerical ones have been found for all dimensions up to 196. Solutions in larger dimensions have been obtained as part of conjectured families obeying additional symmetries, derived from solutions in smaller dimensions.
The talk will highlight the methods used to find numerical and exact SIC-POVMs. Links to number-theoretic conjectures in this context allow to convert numerical solutions into exact solutions.
Further inspection of the exact solutions indicates new connections to Stark's conjectures about the relation between L-functions and certain algebraic units in ray class fields. Using those Stark units, we have been able to compute SIC-POVMs in some four- and five-digit dimensions, the largest being 39604.
The talk is partially based on joint work with with Marcus Appleby, Ingemar Bengtsson, Michael Harrison, Gary McConnell, and Andrew Scott.