Symmetries of Weyl-Heisenberg SIC-POVMs

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, exact
fiducial vectors are known for only a few dimensions, and numerical
ones up to around dimension 150, together with some solutions in
larger dimensions. Recently, exact and numerical solutions in
record-dimensions 323 and 844, respectively, have been found as part
of conjectured families obeying additional symmetries.

The talk will present the symmetries for this putative family together
with generalisations.  Links to number-theoretic conjectures in this
context will be presented as well.  This allows to convert numerical
solutions of moderate precision into exact solutions, including
dimension 844 mentioned above as well as the new record dimension 1299.