Codes and Expansions (CodEx) Seminar

Markus Grassl (International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk)
Computing SIC-POVMs using Permutation Symmetries and Stark Units

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 \(\mathbb{C}^d\) such that the inner product between any pair of vectors has constant modulus.

Zauner conjectured that SIC-POVMs existed for all finite dimensions and that they could 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, albeit 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.

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, including dimensions 844 and 1299.

Further inspection of the exact solutions indicates new connections to Stark's conjectures. Using 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, and Gary McConnell.