Geometric properties of SIC-POVM tensor square. (English) Zbl 1523.47081
Summary: 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 [M. Appleby et al., J. Phys. A, Math. Theor. 52, No. 29, Article ID 295301, 26 p. (2019; Zbl 1509.81010)].
MSC:
| 47N50 | Applications of operator theory in the physical sciences |
| 42C15 | General harmonic expansions, frames |
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
symmetric, informationally complete, positive operator-valued measure (SIC-POVM); Zauner conjecture; Weyl-Heisenberg groupCitations:
Zbl 1509.81010References:
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