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.