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

SIGMA 5 (2009), 096, 15 pages      arXiv:0903.5418

Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Hans Havlicek a, c, Boris Odehnal a and Metod Saniga b, c
a) Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, A-1040 Wien, Austria
b) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
c) Center for Interdisciplineary Research (ZiF), University of Bielefeld, D-33615 Bielefeld, Germany

Received August 19, 2009, in final form October 02, 2009; Published online October 13, 2009

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups. Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G commute. Restricting to p = 2, we search for ''refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of G is ≤ 2. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ''condensation'' of several distinct elements of G. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.

Key words: groups; symplectic and orthogonal polar spaces; geometry of generalised Pauli groups.

pdf (383 kb)   ps (311 kb)   tex (203 kb)


  1. Huppert B., Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967.
  2. Shaw R., Finite geometries and Clifford algebras, J. Math. Phys. 30 (1989), 1971-1984.
  3. Shaw R., Clifford algebras, spinors and finite geometries, in Group Theoretical Methods in Physics (Moscow, 1990), Lecture Notes in Phys., Vol. 382, Springer, Berlin, 1991, 527-530.
  4. Shaw R., Finite geometries and Clifford algebras. III, in Clifford Algebras and Their Applications in Mathematical Physics (Montpellier, 1989), Fund. Theories Phys., Vol. 47, Kluwer Acad. Publ., Dordrecht, 1992, 121-132.
  5. Shaw R., Finite geometry and the table of real Clifford algebras, in Clifford Algebras and Their Applications in Mathematical Physics (Deinze, 1993), Fund. Theories Phys., Vol. 55, Kluwer Acad. Publ., Dordrecht, 1993, 23-31.
  6. Shaw R., Finite geometry, Dirac groups and the table of real Clifford algebras, in Clifford Algebras and Spinor Structures, Math. Appl., Vol. 321, Kluwer Acad. Publ., Dordrecht, 1995, 59-99.
  7. Shaw R., Jarvis T.M., Finite geometries and Clifford algebras. II, J. Math. Phys. 31 (1990), 1315-1324.
  8. Gordon N.A., Jarvis T.M., Maks J.G., Shaw R., Composition algebras and PG(m,2), J. Geom. 51 (1994), 50-59.
  9. Planat M., Saniga M., Kibler M.R., Quantum entanglement and projective ring geometry, SIGMA 2 (2006), 066, 14 pages, quant-ph/0605239.
  10. Saniga M., Planat M., Finite geometries in quantum theory: from Galois (fields) to Hjelmslev (rings), Internat. J. Modern Phys. B 20 (2006), 1885-1892.
  11. Saniga M., Planat M., A projective line over the finite quotient ring GF(2)[x]/<x3x> and quantum entanglement: theoretical background, Theoret. and Math. Phys. 151 (2007), 474-481, quant-ph/0603051.
  12. Saniga M., Planat M., Minarovjech M., Projective line over the finite quotient ring GF(2)[x]/<x3x> and quantum entanglement: the Mermin "magic" square/pentagram, Theoret. and Math. Phys. 151 (2007), 625-631, quant-ph/0603206.
  13. Saniga M., Planat M., Multiple qubits as symplectic polar spaces of order two, Adv. Stud. Theor. Phys. 1 (2007), 1-4, quant-ph/0612179.
  14. Havlicek H., Saniga M., Projective ring line of a specific qudit, J. Phys. A: Math. Theor. 40 (2007), F943-F952, arXiv:0708.4333.
  15. Saniga M., Planat M., Pracna P., Havlicek H., The Veldkamp space of two-qubits, SIGMA 3 (2007), 075, 7 pages, arXiv:0704.0495.
  16. Planat M., Baboin A.-C., Qudits of composite dimension, mutually unbiased bases and projective ring geometry, J. Phys. A: Math. Theor. 40 (2007), F1005-F1012, arXiv:0709.2623.
  17. Planat M., Baboin A.-C., Saniga M., Multi-line geometry of qubit-qutrit and higher-order Pauli operators, Internat. J. Theoret. Phys. 47 (2008), 1127-1135, arXiv:0705.2538.
  18. Havlicek H., Saniga M., Projective ring line of an arbitrary single qudit, J. Phys. A: Math. Theor. 41 (2008), 015302, 12 pages, arXiv:0710.0941.
  19. Planat M., Saniga M., On the Pauli graphs of N-qudits, Quantum Inf. Comput. 8 (2008), 127-146, quant-ph/0701211.
  20. Saniga M., Planat M., Pracna P., Projective ring line encompassing two-qubits, Theoret. and Math. Phys. 155 (2008), 905-913, quant-ph/0611063.
  21. Lévay P., Saniga M., Vrana P., Three-qubit operators, the split Cayley hexagon of order two and black holes, Phys. Rev. D 78 (2008), 124022, 16 pages, arXiv:0808.3849.
  22. Lévay P., Saniga M., Vrana P., Pracna P., Black hole entropy and finite geometry, Phys. Rev. D 79 (2009), 084036, 12 pages, arXiv:0903.0541.
  23. Rau A.R.P., Mapping two-qubit operators onto projective geometries, Phys. Rev. A 79 (2009), 042323, 6 pages, arXiv:0808.0598.
  24. Thas K., Pauli operators of N-qubit Hilbert spaces and the Saniga-Planat conjecture, Chaos, Solitons, Fractals, to appear.
  25. Thas K., The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs, Europhys. Lett. EPL 86 (2009), 60005, 3 pages.
  26. Kirsch A., Beziehungen zwischen der Additivität und der Homogenität von Vektorraum-Abbildungen, Math.-Phys. Semesterber. 25 (1978), 207-210.
  27. Mayr U., Zur Definition der linearen Abbildung, Math.-Phys. Semesterber. 26 (1979), 216-222.
  28. Buekenhout F., Cameron P., Projective and affine geometry over division rings, in Handbook of Incidence Geometry, Editor F. Buekenhout, North-Holland, Amsterdam, 1995, 27-62.
  29. Cameron P.J., Projective and polar spaces, available at
  30. Hirschfeld J.W.P., Projective geometries over finite fields, 2nd ed., Clarendon Press, Oxford, 1998.
  31. Borsten L., Dahanayake D., Duff M.J., Ebrahim H., Rubens W., Black holes, qubits and octonions, Phys. Rep. 471 (2009), 113-219, arXiv:0809.4685.
  32. Payne S.E., Thas J.A., Finite generalized quadrangles, Research Notes in Mathematics, Vol. 110, Pitman (Advanced Publishing Program), Boston, MA, 1984.

Previous article   Next article   Contents of Volume 5 (2009)