• DOI:10.1007/s11005-021-01496-w
• Corpus ID: 207930399

Geometric properties of SIC-POVM tensor square

@article{Ostrovskyi2019GeometricPO,
  title={Geometric properties of SIC-POVM tensor square},
  author={Vasyl Ostrovskyi and Danylo Yakymenko},
  journal={Letters in Mathematical Physics},
  year={2019},
  volume={112},
  pages={1-27}
}

• V. Ostrovskyi, D. Yakymenko
• Published 13 November 2019
• Mathematics
• Letters in Mathematical Physics

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