Numerical Investigations of Symmetric Informationally Complete POVMs

Symmetric informationally complete quantum measurements or SIC-POVMs are interesting from a number of perspectives. For instance, in QBism, where quantum states are understood as degrees belief rather than objective features of nature, SIC-POVMs give a way of representing the Born rule so that the distinction between it and the classical law of total probability is the minimum possible [DeBrota, Fuchs, and Stacey, Phys. Rev. Res. 2, 013074 (2020)]. SIC-POVMs are also optimal for linear quantum-state tomography and a number of other quantum information applications. In the search for SIC-POVMs, we are most interested in the group covariant case, where the problem boils down to finding a single fiducial vector for generating the whole structure. For finite dimensions d, this amounts to finding a solution to d^2 simultaneous fourth-order polynomial equations generated by the discrete Weyl-Heisenberg group. However, it has been conjectured that it is already enough to satisfy only 3d/2 of the defining equations to find a solution [Appleby, Dang, and Fuchs, Entropy 16, 1484 (2014)]. In this talk, I will present the most up-to-date numerical findings with regard to this conjecture.