Skip to main content
SpringerLink
Log in

Discrete Phase-Space Structures and Mutually Unbiased Bases

  • Conference paper
Book cover Arithmetic of Finite Fields (WAIFI 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4547))

Included in the following conference series:

Abstract

We propose a unifying phase-space approach to the construction of mutually unbiased bases for an n-qubit system. It is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional conditions. The effect of local transformations is also studied.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Schwinger, J.: The geometry of quantum states. Proc. Natl. Acad. Sci. USA 46, 257–265 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  2. Wootters, W.K.: A Wigner-function formulation of finite-state quantum mechanics. Ann. Phys (NY) 176, 1–21 (1987)

    Article  MathSciNet  Google Scholar 

  3. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)

    Article  MathSciNet  Google Scholar 

  4. Lawrence, J., Brukner, Č., Zeilinger, A.: Mutually unbiased binary observable sets on N qubits. Phys. Rev. A 65, 32320 (2002)

    Article  Google Scholar 

  5. Chaturvedi, S.: Aspects of mutually unbiased bases in odd-prime-power dimensions. Phys. Rev. A 65, 44301 (2002)

    Article  Google Scholar 

  6. Wootters, W.K.: Quantum measurements and finite geometry. Found. Phys. 36, 112–126 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys (NY) 191, 363–381 (1989)

    Article  MathSciNet  Google Scholar 

  8. Asplund, R., Björk, G.: Reconstructing the discrete Wigner function and some properties of the measurement bases. Phys. Rev. A 64, 12106 (2001)

    Article  Google Scholar 

  9. Bechmann-Pasquinucci, H., Peres, A.: Quantum cryptography with 3-State systems. Phys. Rev. Lett. 85, 3313–3316 (2000)

    Article  MathSciNet  Google Scholar 

  10. Cerf, N., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using d-level systems. Phys. Rev. A 88, 127902 (2002)

    Article  Google Scholar 

  11. Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862–1868 (1996)

    Article  MathSciNet  Google Scholar 

  12. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78, 405–408 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Vaidman, L., Aharonov, Y., Albert, D.Z.: How to ascertain the values of σ x , σ y , and σ z of a spin-1/2 particle. Phys. Rev. Lett. 58, 1385–1387 (1987)

    Article  MathSciNet  Google Scholar 

  14. Englert, B.-G., Aharonov, Y.: The mean king’s problem: prime degrees of freedom. Phys. Lett. A 284, 1–5 (2001)

    Article  MathSciNet  Google Scholar 

  15. Aravind, P.K.: Solution to the king’s problem in prime power dimensions. Z. Naturforsch. A. Phys. Sci. 58, 85–92 (2003)

    Google Scholar 

  16. Schulz, O., Steinhübl, R., Weber, M., Englert, B.-G, Kurtsiefer, C., Weinfurter, H.: Ascertaining the values of σ x , σ y , and σ z of a polarization qubit. Phys. Rev. Lett. 90, 177901 (2003)

    Article  Google Scholar 

  17. Kimura, G., Tanaka, H., Ozawa, M.: Solution to the mean king’s problem with mutually unbiased bases for arbitrary levels. Phys. Rev. A 73, 50301 (R) (2006)

    Article  Google Scholar 

  18. Ivanović, I.D.: Geometrical description of quantal state determination. J. Phys. A 14, 3241–3246 (1981)

    Article  MathSciNet  Google Scholar 

  19. Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: ℤ4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. London Math. Soc. 75, 436–480 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, V.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Klappenecker, A., Rötteler, M.: Constructions of mutually unbiased bases. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) Finite Fields and Applications. LNCS, vol. 2948, pp. 137–144. Springer, Heidelberg (2004)

    Google Scholar 

  22. Lawrence, J.: Mutually unbiased bases and trinary operator sets for N qutrits. Phys. Rev. A 70, 12302 (2004)

    Article  MathSciNet  Google Scholar 

  23. Parthasarathy, K.R.: On estimating the state of a finite level quantum system. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 607–617 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pittenger, A.O., Rubin, M.H.: Wigner function and separability for finite systems. J. Phys. A 38, 6005–6036 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Durt, T.: About mutually unbiased bases in even and odd prime power dimensions. J. Phys. A 38, 5267–5284 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Planat, M., Rosu, H.: Mutually unbiased phase states, phase uncertainties, and Gauss sums. Eur. Phys. J. D 36, 133–139 (2005)

    Article  Google Scholar 

  27. Klimov, A.B., Sánchez-Soto, L.L., de Guise, H.: Multicomplementary operators via finite Fourier transform. J. Phys. A 38, 2747–2760 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge (1986)

    MATH  Google Scholar 

  29. Buot, F.A.: Method for calculating Tr \(\mathcal{H^{n}}\) in solid-state theory. Phys. Rev. B 10, 3700–3705 (1974)

    Article  Google Scholar 

  30. Galetti, D., De Toledo Piza, A.F.R.: An extended Weyl-Wigner transformation for special finite spaces. Physica A 149, 267–282 (1988)

    Article  MathSciNet  Google Scholar 

  31. Cohendet, O., Combe, P., Sirugue, M., Sirugue-Collin, M.: A stochastic treatment of the dynamics of an integer spin. J. Phys. A 21, 2875–2884 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  32. Wootters, W.K.: Picturing qubits in phase space. IBM J. Res. Dev. 48, 99–110 (2004)

    Article  Google Scholar 

  33. Gibbons, K.S., Hoffman, M.J., Wootters, W.K.: Discrete phase space based on finite fields. Phys. Rev. A 70, 62101 (2004)

    Article  MathSciNet  Google Scholar 

  34. Paz, J.P., Roncaglia, A.J., Saraceno, M.: Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem. Phys. Rev. A 72, 12309 (2005)

    Article  Google Scholar 

  35. Durt, T.: About Weyl and Wigner tomography in finite-dimensional Hilbert spaces. Open Syst. Inf. Dyn. 13, 403–413 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Klimov, A.B., Munoz, C., Romero, J.L.: Geometrical approach to the discrete Wigner function in prime power dimensions. J. Phys. A 39, 14471–14497 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Romero, J.L., Björk, G., Klimov, A.B., Sánchez-Soto, L.L.: On the structure of the sets of mutually unbiased bases for N qubits. Phys. Rev. A 72, 62310 (2005)

    Article  Google Scholar 

  38. Vourdas, A.: Quantum systems with finite Hilbert space. Rep. Prog. Phys. 67, 267–320 (2004)

    Article  MathSciNet  Google Scholar 

  39. Englert, B.-G., Metwally, N.: Separability of entangled q-bit pairs. J. Mod. Opt. 47, 2221–2231 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  40. 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, 371–379 (2007)

    Article  Google Scholar 

  41. Klimov, A.B., Romero, J.L., Björk, G., Sánchez-Soto, L.L.: J. Phys. A 40, 3987–3998 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física, Universidad de Guadalajara, Revolución 1500, 44420 Guadalajara, Jalisco, Mexico

    A. B. Klimov & J. L. Romero

  2. School of Information and Communication Technology, Royal Institute of Technology (KTH), Electrum 229, SE-164 40 Kista, Sweden

    G. Björk

  3. Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain

    L. L. Sánchez-Soto

Authors
  1. A. B. Klimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. L. Romero
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Björk
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. L. Sánchez-Soto
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Claude Carlet Berk Sunar

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Klimov, A.B., Romero, J.L., Björk, G., Sánchez-Soto, L.L. (2007). Discrete Phase-Space Structures and Mutually Unbiased Bases. In: Carlet, C., Sunar, B. (eds) Arithmetic of Finite Fields. WAIFI 2007. Lecture Notes in Computer Science, vol 4547. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73074-3_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73074-3_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73073-6

  • Online ISBN: 978-3-540-73074-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation