Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 044, 12 pages      arXiv:2312.13400      https://doi.org/10.3842/SIGMA.2024.044

SICs and the Triangle Group $(3,3,3)$

Danylo Yakymenko
Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine

Received January 02, 2024, in final form May 10, 2024; Published online May 29, 2024

Abstract
The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group $(3,3,3)$. We give a full description of such representations and show how it can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension $d>3$ is prime.

Key words: quantum design; SIC-POVM; equiangular tight frame; Zauner's conjecture; Weyl-Heisenberg group; triangle group; projective representation.

pdf (579 kb)   tex (187 kb)  

References

  1. Andersson O., Dumitru I., Aligned SICs and embedded tight frames in even dimensions, J. Phys. A 52 (2019), 425302, 25 pages, arXiv:1905.09737.
  2. Appleby D.M., Symmetric informationally complete-positive operator valued measures and the extended Clifford group, J. Math. Phys. 46 (2005), 052107, 29 pages, arXiv:quant-ph/0412001.
  3. Appleby M., Bengtsson I., Dumitru I., Flammia S., Dimension towers of SICs. I. Aligned SICs and embedded tight frames, J. Math. Phys. 58 (2017), 112201, 19 pages, arXiv:1707.09911.
  4. Appleby M., Bengtsson I., Grassl M., Harrison M., McConnell G., SIC-POVMs from Stark units: prime dimensions $n^2 + 3$, J. Math. Phys. 63 (2022), 112205, 31 pages, arXiv:2112.05552.
  5. Appleby M., Bengtsson I., Grassl M., Harrison M., SIC-POVMs from Stark Units: Dimensions $n^2+3=4p$, $p$ prime, (2024), arXiv:2403.02872.
  6. Appleby M., Flammia S., McConnell G., Yard J., SICs and algebraic number theory, Found. Phys. 47 (2017), 1042-1059, arXiv:1701.05200.
  7. Bengtsson I., Blanchfield K., Campbell E., Howard M., Order 3 symmetry in the Clifford hierarchy, J. Phys. A 47 (2014), 455302, 13 pages, arXiv:1407.2713.
  8. Bos L., Waldron S., SICs and the elements of order three in the Clifford group, J. Phys. A 52 (2019), 105301, 31 pages.
  9. Davidson K.R., $C^*$-algebras by example, Fields Inst. Monogr., Vol. 6, American Mathematical Society, Providence, RI, 1996.
  10. Flammia S.T., On SIC-POVMs in prime dimensions, J. Phys. A 39 (2006), 13483-13493, arXiv:quant-ph/0605050.
  11. Fuchs C., Hoang M., Stacey B., The SIC Question: History and State of Play, Axioms 6 (2017), 21, 20 pages, arXiv:1703.07901.
  12. Kirillov A.A., Elements of the theory of representations, Grundlehren Math. Wiss., Vol. 220, Springer, Berlin, 1976.
  13. Kopp G.S., SIC-POVMs and the Stark conjectures, Int. Math. Res. Not. 2021 (2021), 13812-13838, arXiv:1807.05877.
  14. Livins'kyi I.V., Radchenko D.V., Representations of algebras defined by a multiplicative relation and corresponding to the extended Dynkin graphs $\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$, Ukrainian Math. J. 64 (2013), 1865-1892.
  15. Magnus W., Noneuclidean tesselations and their groups, Pure Appl. Math., Vol. 61, Academic Press, New York, 1974.
  16. Renes J.M., Blume-Kohout R., Scott A.J., Caves C.M., Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004), 2171-2180, arXiv:quant-ph/0310075.
  17. Scott A.J., SICs: Extending the list of solutions, arXiv:1703.03993.
  18. Scott A.J., Tight informationally complete quantum measurements, J. Phys. A 39 (2006), 13507-13530, arXiv:quant-ph/0604049.
  19. Scott A.J., Grassl M., Symmetric informationally complete positive-operator-valued measures: a new computer study, J. Math. Phys. 51 (2010), 042203, 16 pages, arXiv:0910.5784.
  20. Weyl H., The theory of groups and quantum mechanics, Dover Publications, New York, 1950.
  21. Zauner G., Quantum designs: foundations of a noncommutative design theory, Int. J. Quantum Inf. 9 (2011), 445-507.
  22. Zhu H., SIC POVMs and Clifford groups in prime dimensions, J. Phys. A 43 (2010), 305305, 24 pages, arXiv:1003.3591.

Previous article  Next article  Contents of Volume 20 (2024)