Discrete phase-space structures and Wigner functions for N qubits

  • C. Muñoz
  • A. B. Klimov
  • L. Sánchez-Soto


We further elaborate on a phase-space picture for a system of N qubits and explore the structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves satisfying certain additional properties and different entanglement properties. We discuss the construction of discrete covariant Wigner functions for these bundles and provide several illuminating examples.


Wigner function Mutually unbiased bases keyword Entanglement 



This work is partially supported by the Grant 254127 of CONACyT (Mexico). L. L. S. S. acknowledges the support of the Spanish MINECO (Grant FIS2015-67963-P).


  1. 1.
    Schroek, F.E.: Quantum Mechanics on Phase Space. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  2. 2.
    Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, Berlin (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Zachos, C.K., Fairlie, D.B., Curtright, T.L. (eds.): Quantum Mechanics in Phase Space. World Scientific, Singapore (2005)zbMATHGoogle Scholar
  4. 4.
    Weyl, H.: Gruppentheorie und Quantemechanik. Hirzel-Verlag, Leipzig (1928)zbMATHGoogle Scholar
  5. 5.
    Hannay, J.H., Berry, M.V.: Quantization of linear maps on a Torus–Fresnel diffraction by aperiodic grating. Phys. D 1, 267–290 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Leonhardt, U.: Quantum-state tomography and discrete Wigner function. Phys. Rev. Lett. 74, 4101–4103 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zak, J.: Doubling feature of the Wigner function: finite phase space. J. Phys. A Math. Theor. 44, 345305 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Miquel, C., Paz, J.P., Saraceno, M.: Quantum computers in phase space. Phys. Rev. A 65, 062309 (2002)ADSCrossRefGoogle Scholar
  9. 9.
    Paz, J.P.: Discrete Wigner functions and the phase-space representation of quantum teleportation. Phys. Rev. A 65, 062311 (2002)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Buot, F.A.: Method for calculating \({{\rm Tr}} {\cal{H}}^n\) in solid-state theory. Phys. Rev. B 10, 3700–3705 (1973)ADSCrossRefGoogle Scholar
  11. 11.
    Schwinger, J.: Unitary operator basis. Proc. Natl. Acad. Sci. USA 46, 570–576 (1960)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Galetti, D., de Toledo Piza, A.F.R.: An extended Weyl–Wigner transformation for special finite spaces. Phys. A 149, 267–282 (1988)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cohendet, O., Combe, P., Sirugue, M., Sirugue-Collin, M.: A stochastic treatment of the dynamics of an integer spin. J. Phys. A Math. Gen. 21, 2875–2884 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Galetti, D., de Toledo Piza, A.F.R.: Discrete quantum phase spaces and the mod \(N\) invariance. Phys. A 186, 513–523 (1992)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kasperkovitz, P., Peev, M.: Wigner–Weyl formalism for toroidal geometries Ann. Phys. 230, 21–51 (1994)zbMATHGoogle Scholar
  17. 17.
    Rivas, A.M.F., de Almeida, A.M.O.: The Weyl representation on the torus. Ann. Phys. 276, 223–256 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Vourdas, A.: Quantum systems with finite Hilbert space. Rep. Prog. Phys. 67, 267–320 (2004)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Ligabò, M.: Torus as phase space: Weyl quantization, dequantization, and Wigner formalism. J. Math. Phys. 57, 082110 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wootters, W.K.: A Wigner-function formulation of finite-state quantum mechanics. Ann. Phys. 176, 1–21 (1987)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gibbons, K.S., Hoffman, M.J., Wootters, W.K.: Discrete phase space based on finite fields. Phys. Rev. A 70, 062101 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chuang, I., Nielsen, M.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  23. 23.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)zbMATHGoogle Scholar
  24. 24.
    Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys. 16, 391–405 (1986)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Romero, J.L., Björk, G., Klimov, A.B., Sánchez-Soto, L.L.: Structure of the sets of mutually unbiased bases for \(n\) qubits. Phys. Rev. A 72, 062310 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    Björk, G., Romero, J.L., Klimov, A.B., Sánchez-Soto, L.L.: Mutually unbiased bases and discrete Wigner functions. J. Opt. Soc. Am. B 24, 371–378 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Galvão, E.F.: Discrete Wigner functions and quantum computational speedup. Phys. Rev. A 71, 042302 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    Cormick, C., Galvão, E.F., Gottesman, D., Paz, J.P., Pittenger, A.O.: Interference in discrete Wigner functions. Phys. Rev. A 74, 062315 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    Gross, D.: Hudson’s theorem for finite-dimensional quantum systems. J. Math. Phys. 47, 122107 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Klimov, A.B., Sánchez-Soto, L.L., de Guise, H.: Multicomplementary operators via finite Fourier transform. J. Phys. A 38, 2747–2760 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wootters, W.K.: Picturing qubits in phase space. IBM J. Res. Dev. 48, 99–110 (2004)CrossRefGoogle Scholar
  33. 33.
    Klimov, A.B., Romero, J.L., Björk, G., Sánchez-Soto, L.L.: Discrete phase-space structure of \(n\)-qubit mutually unbiased bases. Ann. Phys. 324, 53–72 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Klimov, A.B., Muñoz, C., Romero, J.L.: Geometrical approach to the discrete Wigner function in prime power dimensions. J. Phys. A 39, 14471–14480 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Duncan, R., Perdrix, S.: chap. Graph states and the necessity of Euler decomposition. In: Mathematical Theory and Computational Practice, pp. 167–177. Springer, Berlin (2009)Google Scholar
  37. 37.
    Björk, G., Klimov, A.B., Sánchez-Soto, L.L.: The discrete Wigner function. Prog. Opt. 51, 469–516 (2008)CrossRefGoogle Scholar
  38. 38.
    Hudson, R.L.: When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249–252 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Soto, F., Claverie, P.: When is the Wigner function of multidimensional systems nonnegative? J. Math. Phys. 24, 97–100 (1983)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Kenfack, A., Życzkowski, K.: Negativity of the Wigner function as an indicator of non-classicality. J. Opt. B Quantum Semiclass. Opt. 6, 396–404 (2004)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Siyouri, F., El Baz, M., Hassouni, Y.: The negativity of Wigner function as a measure of quantum correlations. Quantum Inf. Proc. 15, 4237–4252 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Weigert, S., Wilkinson, M.: Mutually unbiased bases for continuous variables. Phys. Rev. A 78, 020303(R) (2008)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidad de GuadalajaraGuadalajaraMexico
  2. 2.Departamento de Óptica, Facultad de FísicaUniversidad ComplutenseMadridSpain
  3. 3.Max-Planck-Institut für die Physik des LichtsErlangenGermany

Personalised recommendations