Geometric properties of SIC-POVM tensor square

It is known that if $$d^2$$ vectors in a d-dimensional Hilbert space H form symmetric, informationally complete, positive operator-valued measure (SIC-POVM), then the tensor squares of these an equiangular tight frame symmetric subspace $$H\otimes H$$ . We prove that, for any SIC-POVM Weyl–Heisenberg group covariant type (WH-type), this can be obtained by projecting WH-type basis onto subspace. give full description set all bases, so could used as search finding SIC solutions. Also, we show particular element close to solution some structural sense. Finally, geometric construction SIC-related fusion frames were discovered odd dimensions Appleby et al. (J Phys A 52(29):295301, 2019).