Group Theoretical Classification of SIC-POVMs

The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions $\leq 151$ and few higher dimensions as high as $1155$. All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3 it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint. We show analytically that the SIC-POVM Gram matrices exist on critical points of surfaces formed by the two functions on a subspace of symmetric matrices and we show numerically that in dimensions 4 to 7, all SIC-POVM Gram matrices lie in disjoint solution "islands". We generate $O(10^6)$ and $O(10^5)$ Gram matrices in dimensions 4 and 5, respectively and $O(10^2)$ Gram matrices in dimensions 6 and 7. For every Gram matrix obtained, we generate the symmetry groups and show that all symmetry groups contain a subgroup of $3n^2$ elements. The elements of the subgroup correspond to the Weyl-Heisenberg group matrices and the order-3 unitaries that generate them. All constructed Gram matrices have a unique generating set. Using this fact, we generate permutation matrices to map the Gram matrices to known Weyl-Heisenberg group covariant solutions. In dimensions 4 and 5, the absence of a solution with a smaller symmetry, strongly suggests that non-group covariant SIC-POVMs cannot be constructed.