Abstract
It is known that if \(d^2\) 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 \(H\otimes H\). 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 \(H\otimes H\) 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).
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Ostrovskyi, V., Yakymenko, D. Geometric properties of SIC-POVM tensor square. Lett Math Phys 112, 7 (2022). https://doi.org/10.1007/s11005-021-01496-w
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DOI: https://doi.org/10.1007/s11005-021-01496-w