Tight frames, Hadamard matrices and Zauner’s conjecture. (English) Zbl 1509.81010
Summary: 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 \(d(d\pm1)/2\). 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 are non-linear. Restricted defect calculations are 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 \(d\geqslant3\).
MSC:
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 15B34 | Boolean and Hadamard matrices |
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