Abstract

We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and complex Hadamard matrix in dimension d(2), and a pair of ETFs (equiangular tight frames) in dimensions d(d +/- 1)/2. We also show that a WH (Weyl-Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d. We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds arc non-linear. Restricted defect calculations arc consistent with this being true for the structures associated to every known SIC with d between 3 and 16, suggesting it may be true for all d >= 3.