SIGMA 3 (2007), 075, 7 pages      arXiv:0704.0495      https://doi.org/10.3842/SIGMA.2007.075

The Veldkamp Space of Two-Qubits

Metod Saniga ^a, Michel Planat ^b, Petr Pracna ^c and Hans Havlicek ^d
a) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
^b) Institut FEMTO-ST, CNRS, Département LPMO, 32 Avenue de l'Observatoire, F-25044 Besançon Cedex, France
^c) J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejskova 3, CZ-182 23 Prague 8, Czech Republic
^d) Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria

Received April 13, 2007, in final form June 18, 2007; Published online June 29, 2007

Abstract
Given a remarkable representation of the generalized Pauli operators of two-qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements - the so-called Veldkamp space of W(2). An intriguing novelty is the recognition of (uni- and tri-centric) triads and specific pentads of the Pauli operators in addition to the ''classical'' subsets answering to geometric hyperplanes of W(2).

Key words: generalized quadrangles; Veldkamp spaces; Pauli operators of two-qubits.

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References

1. Saniga M., Planat M., Pracna P., Projective ring line encompassing two-qubits, Theor. Math. Phys., to appear, quant-ph/0611063.
2. Saniga M., Planat M., Multiple qubits and symplectic polar spaces of order two, Adv. Studies Theor. Phys. 1 (2007) 1-4, quant-ph/0612179.
3. Planat M., Saniga M., On the Pauli graph of N-qudits, quant-ph/0701211.
4. Payne S.E., Thas J.A., Finite generalized quadrangles, Pitman, Boston - London - Melbourne, 1984.
5. Buekenhout F., Cohen A.M., Diagram geometry (Chapter 10.2), Springer, New York, to appear; preprints of separate chapters can be found at http://www.win.tue.nl/~amc/buek/.
6. Ronan M.A., Embeddings and hyperplanes of discrete geometries, European J. Combin. 8 (1987), 179-185.
7. Shult E., On Veldkamp lines, Bull. Belg. Math. Soc. 4 (1997), 299-316.
8. Hirschfeld J.W.P., Thas J.A., General Galois geometries, Oxford University Press, Oxford, 1991.
9. Lawrence J., Brukner C., Zeilinger A., Mutually unbiased binary observable sets on N qubits, Phys. Rev. A 65 (2002), 032320, 5 pages, quant-ph/0104012.
10. Saniga M., Planat M., Rosu H., Mutually unbiased bases and finite projective planes, J. Opt. B: Quantum Semiclass. Opt. 6 (2004), L19-L20, math-ph/0403057.
11. Saniga M., Planat M., Minarovjech M., Projective line over the finite quotient ring GF(2)[x]/(x³ - x) and quantum entanglement: the Mermin "magic" square/pentagram, Theor. Math. Phys. 151 (2007), 625-631, quant-ph/0603206.
12. Planat M., Saniga M., Pauli graph and finite projective lines/geometries, Optics and Optoelectronics, to appear, quant-ph/0703154.
13. Thas K., Pauli operators of N-qubit Hilbert spaces and the Saniga-Planat conjecture, Chaos Solitons Fractals, to appear.
14. Thas K., The geometry of generalized Pauli operators of N-qudit Hilbert space, Quantum Information and Computation, submitted.