Markus Grassl: Computing Numerical and Exact SIC-POVMs

Chaos and quantum chaos seminar at the Institute of Physics, Jagiellonian University.

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 normalised vectors in the d-dimensional complex vector space such that the inner product between any pair of vectors has constant modulus. It has been conjectured that SIC-POVMs exist for all dimensions and that they can be constructed as orbits of a so-called fiducial vector under the Weyl-Heisenberg group. Despite a lot of effort, numerical or exact fiducial vectors are only known for a finite, though growing list of dimensions. Currently, numerical ones have been found for all dimensions up to 193. Solutions in larger dimensions have been found as part of conjectured families obeying additional symmetries. After an introduction to SIC-POVMs, 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 of moderate precision into exact solutions, including dimensions 844 and 1299. We will also briefly address recent results obtained in collaboration with Marcus Appleby, Ingemar Bengtsson, Michael Harrison, and Gary McConnell.