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SIGMA, 2007, Volume 3, 075, 7 pages (Mi sigma201)  

This article is cited in 22 scientific papers (total in 22 papers)

The Veldkamp Space of Two-Qubits

Metod Sanigaa, Michel Planatb, Petr Pracnac, Hans Havlicekd

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

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 $W(2)$.

Keywords: generalized quadrangles; Veldkamp spaces; Pauli operators of two-qubits


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ArXiv: 0704.0495
MSC: 51Exx; 81R99
Received: April 13, 2007; in final form June 18, 2007; Published online June 29, 2007

Citation: Metod Saniga, Michel Planat, Petr Pracna, Hans Havlicek, “The Veldkamp Space of Two-Qubits”, SIGMA, 3 (2007), 075, 7 pp.

Citation in format AMSBIB
\by Metod Saniga, Michel Planat, Petr Pracna, Hans Havlicek
\paper The Veldkamp Space of Two-Qubits
\jour SIGMA
\yr 2007
\vol 3
\papernumber 075
\totalpages 7

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    This publication is cited in the following articles:
    1. Metod Saniga, Hans Havlicek, Michel Planat, Petr Pracna, “Twin “Fano-Snowflakes” over the Smallest Ring of Ternions”, SIGMA, 4 (2008), 050, 7 pp.  mathnet  crossref  mathscinet  zmath
    2. Kibler M.R., “Variations on a theme of Heisenberg, Pauli and Weyl”, J. Phys. A, 41:37 (2008), 375302, 19 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    3. Planat M., Baboin A.-., Saniga M., “Multi-line geometry of qubit-qutrit and higher-order Pauli operators”, Internat. J. Theoret. Phys., 47:4 (2008), 1127–1135  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Kibler M.R., “An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group”, J. Phys. A, 42:35 (2009), 353001, 28 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    5. Lévay P., Saniga M., Vrana P., Pracna P., “Black hole entropy and finite geometry”, Phys. Rev. D, 79:8 (2009), 084036, 12 pp.  crossref  mathscinet  adsnasa  isi  scopus
    6. Rau A.R.P., “Mapping two-qubit operators onto projective geometries”, Phys. Rev. A (3), 79:4 (2009), 042323, 6 pp.  crossref  mathscinet  adsnasa  isi  scopus
    7. Hans Havlicek, Boris Odehnal, Metod Saniga, “Factor-Group-Generated Polar Spaces and (Multi-)Qudits”, SIGMA, 5 (2009), 096, 15 pp.  mathnet  crossref  mathscinet
    8. Vrana P., Lévay P., “The Veldkamp space of multiple qubits”, J. Phys. A, 43:12 (2010), 125303, 16 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Saniga M., Lévay P., Planat M., Pracna P., “Geometric hyperplanes of the near hexagon $L_3\times GQ(2,2)$”, Lett. Math. Phys., 91:1 (2010), 93–104  crossref  mathscinet  zmath  isi  scopus
    10. Saniga M., Green R.M., Levay P., Vrana P., Pracna P., “THE VELDKAMP SPACE OF GQ(2,4)”, Int J Geom Methods Mod Phys, 7:7 (2010), 1133–1145  crossref  mathscinet  zmath  isi  scopus
    11. Scharnhorst K., van Holten J.-W., “Nonlinear Bogolyubov-Valatin transformations: Two modes”, Ann Physics, 326:11 (2011), 2868–2933  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    12. Metod Saniga, Michel Planat, Petr Pracna, Péter Lévay, “‘Magic’ configurations of three-qubit observables and geometric hyperplanes of the smallest split Cayley hexagon”, SIGMA, 8 (2012), 083, 9 pp.  mathnet  crossref  mathscinet
    13. Green R.M. Saniga M., “The Veldkamp Space of the Smallest Slim Dense Near Hexagon”, Int. J. Geom. Methods Mod. Phys., 10:2 (2013), 1250082  crossref  mathscinet  zmath  isi  scopus
    14. Saniga M., Holweck F., Pracna P., “From Cayley-Dickson Algebras To Combinatorial Grassmannians”, 3, no. 4, 2015, 1192–1221  crossref  zmath  isi
    15. Planat M. Giorgetti A. Holweck F. Saniga M., “Quantum Contextual Finite Geometries From Dessins D'Enfants”, Int. J. Geom. Methods Mod. Phys., 12:7 (2015), 1550067  crossref  mathscinet  zmath  isi  scopus
    16. Saniga M. Havlicek H. Holweck F. Planat M. Pracna P., “Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties”, Ann. Inst. Henri Poincare D, 2:3 (2015), 309–333  crossref  mathscinet  zmath  isi  scopus
    17. Levay P., Szabo Z., “Mermin pentagrams arising from Veldkamp lines for three qubits”, J. Phys. A-Math. Theor., 50:9 (2017), 095201  crossref  mathscinet  zmath  isi  scopus
    18. Saniga M., “A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line”, Entropy, 19:10 (2017), 556  crossref  isi  scopus
    19. Saniga M. Holweck F. Pracna P., “Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups”, Int. J. Geom. Methods Mod. Phys., 14:5 (2017), 1750080  crossref  mathscinet  zmath  isi  scopus
    20. Green R.M., Saniga M., “A Classification of the Veldkamp Lines of the Near Hexagon l-3 X Gq(2,2)”, ARS Math. Contemp., 12:2 (2017), 287–299  mathscinet  zmath  isi
    21. Levay P., Holweck F., Saniga M., “Magic Three-Qubit Veldkamp Line: a Finite Geometric Underpinning For Form Theories of Gravity and Black Hole Entropy”, Phys. Rev. D, 96:2 (2017), 026018  crossref  mathscinet  isi  scopus
    22. Levay P. Holweck F., “Finite Geometric Toy Model of Spacetime as An Error Correcting Code”, Phys. Rev. D, 99:8 (2019), 086015  crossref  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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