On Weyl-Heisenberg orbits of equiangular lines
Mahdad Khatirinejad
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
DOI: 10.1007/s10801-007-0104-1
Abstract
An element z Ĩ \mathbb CP d -1 \mathbf {z}\in \mathbb {CP}^{d-1} is called fiducial if { gz: g\in G} is a set of lines with only one angle between each pair, where G \cong \Bbb Z d \times \Bbb Z d is the one-dimensional finite Weyl-Heisenberg group modulo its centre. We give a new characterization of fiducial vectors. Using this characterization, we show that the existence of almost flat fiducial vectors implies the existence of certain cyclic difference sets. We also prove that the construction of fiducial vectors in prime dimensions 7 and 19 due to Appleby (J. Math. Phys. 46(5):052107, 2005) does not generalize to other prime dimensions (except for possibly a set with density zero). Finally, we use our new characterization to construct fiducial vectors in dimension 7 and 19 whose coordinates are real.
Pages: 333–349
Keywords: keywords complex equiangular lines; Weyl-Heisenberg group; fiducial vector; SIC-POVM
Full Text: PDF
References
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2. Appleby, D.M.: Symmetric informationally complete-positive operator valued measures and the extended Clifford group. J. Math. Phys. 46(5), 052107 (2005)
3. Appleby, D.M., Dang, H.B., Fuchs, C.A.: Physical significance of symmetric informationallycomplete sets of quantum states. Preprint arXiv:0707.2071v1 (2007)
4. Baumert, L.D., Gordon, D.M.: On the existence of cyclic difference sets with small parameters. In: High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams. Fields Inst. Commun., vol. 41, pp. 61-68. Amer. Math. Soc., Providence (2004)
5. Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de finetti representation. J. Math. Phys. 43, 4537-4559 (2002)
6. Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6(3), 363-388 (1977)
7. Flammia, S.T.: On SIC-POVMs in prime dimensions. J. Phys. A 39(43), 13483-13493 (2006)