Advertisement

Qubit representation of qudit states: correlations and state reconstruction

  • Julio A. López-SaldívarEmail author
  • Octavio Castaños
  • Margarita A. Man’ko
  • Vladimir I. Man’ko
Article
  • 32 Downloads

Abstract

A method to establish a qubit decomposition of a general qudit state is presented. This new representation allows a geometrical depiction of any qudit state in the Bloch sphere. Additionally, we show that the nonnegativity conditions of the qudit state imply the existence of quantum correlations between the qubits which compose it. These correlations are used to define new inequalities which the density matrices components must satisfy. The importance of such inequalities in the reconstruction of a qudit state is addressed. As an example of the general procedure, the qubit decomposition of a qutrit system is shown, which allows a classification of the qutrit states by fixing their invariants \(\mathrm{Tr}(\hat{\rho }^2)\), \(\mathrm{Tr}(\hat{\rho }^3)\).

Keywords

Qudit states Geometrical representation of quantum states Quantum correlations Quantum state reconstruction Bloch sphere 

Notes

Acknowledgements

This work was partially supported by DGAPA-UNAM (Under Project IN101619).

References

  1. 1.
    Chernega, V.N., Man’ko, O.V., Man’ko, V.I.: Triangle geometry of the qubit state in the probability representation expressed in terms of the Triada of Malevichs Squares. J. Russ. Laser Res. 38, 141 (2017)CrossRefGoogle Scholar
  2. 2.
    Chernega, V.N., Man’ko, O.V., Man’ko, V.I.: Probability representation of quantum observables and quantum states. J. Russ. Laser Res. 38, 324 (2017)CrossRefGoogle Scholar
  3. 3.
    Chernega, V.N., Man’ko, O.V., Man’ko, V.I.: Triangle geometry for qutrit states in the probability representation. J. Russ. Laser Res. 38, 416 (2017)CrossRefGoogle Scholar
  4. 4.
    López-Saldívar, J.A., Castaños, O., Nahmad-Achar, E., López-Peña, R., Man’ko, V.I., Man’ko, M.A.: Geometry and entanglement of two-qubit states in the quantum probabilistic representation. Entropy 20, 630 (2018)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kurzyński, P., Kołodziejski, A., Laskowski, W., Markiewicz, M.: Three-dimensional visualization of a qutrit. Phys. Rev. A 93, 062126 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    Goyal, S.K., Simon, B.N., Singh, R., Simon, S.: Geometry of the generalized Bloch sphere for qutrits. J. Phys. A 49, 165203 (2016)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kimura, G., Kossakowski, A.: The Bloch-vector space for N-level systems—the spherical-coordinate point of view. Open Syst. Inf. Dyn. 12, 207 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mendaš, I.P.: The classification of three-parameter density matrices for a qutrit. J. Phys. A 39, 11313 (2006)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States, p. 466. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  10. 10.
    Dogra, S., Dorai, K., Arvind.: Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates. J. Phys. B 51, 045505 (2018)Google Scholar
  11. 11.
    Weigert, S.: Pauli problem for a spin of arbitrary length: a simple method to determine its wave function. Phys. Rev. A 45, 7688 (1992)ADSCrossRefGoogle Scholar
  12. 12.
    Buzek, V., Drobny, G., Derka, R., Adam, G., Wiedemann, H.: Quantum state reconstruction from incomplete data. arXiv:quant-ph/9805020 (1998)
  13. 13.
    Collins, D., Gisin, N., Linden, N., Massar, S., Popescu, S.: Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88, 040404 (2002)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ali-Khan, I., Broadbent, C.J., Howell, J.C.: Large-alphabet quantum key distribution using energy-time entangled bipartite states. Phys. Rev. Lett. 98, 060503 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Lloyd, S.: Enhanced sensitivity of photodetection via quantum illumination. Science 321, 1463 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Neeley, M., et al.: Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    Araneda, G., Cisternas, N., Delgado, A.: Telecloning of qudits via partially entangled states. Quantum Inf. Process. 15, 3443 (2016)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Maziero, J.: HilbertSchmidt quantum coherence in multi-qudit systems. Quantum Inf. Process. 16, 274 (2017)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Gedik, Z., et al.: Computational speed-up with a single qudit. Sci. Rep. 5, 14671 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Kues, M., et al.: On-chip generation of high-dimensional entangled quantum states and their coherent control. Nature 546, 622 (2017)ADSCrossRefGoogle Scholar
  21. 21.
    Niu, M.Y., Chuang, I.L., Shapiro, J.H.: Qudit-basis universal quantum computation using \(\chi ^{(2)}\) interactions. Phys. Rev. Lett. 120, 160502 (2018)Google Scholar
  22. 22.
    Lanyon, B.P., et al.: Simplifying quantum logic using higher-dimensional Hilbert spaces. Nat. Phys. 5, 134 (2009)CrossRefGoogle Scholar
  23. 23.
    Ha, D., Kwon, Y.: A minimal set of measurements for qudit-state tomography based on unambiguous discrimination. Quantum Inf. Process. 17, 232 (2018)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Gell-Mann, M.: Symmetries of Baryons and Mesons. Phys. Rev. 125, 1067 (1962)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315 (1994)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Renyi, A.: Probability Theory, p. 672. Dover Publications Inc., New York (2012)zbMATHGoogle Scholar
  27. 27.
    Tsallis, C.: Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status. Springer, Berlin (2001)zbMATHGoogle Scholar
  28. 28.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics, p. 328. Princeton University Press, Princeton (1955)Google Scholar
  29. 29.
    Pearson, K.: VII. Note on regression and inheritance in the case of two parents. Proc. R. Soc. Lond. 58, 240 (1895).  https://doi.org/10.1098/rspl.1895.00412053-9126 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Julio A. López-Saldívar
    • 1
    • 2
    Email author
  • Octavio Castaños
    • 1
  • Margarita A. Man’ko
    • 3
  • Vladimir I. Man’ko
    • 2
    • 3
    • 4
  1. 1.Instituto de Ciencias NuclearesUniversidad Nacional Autnoma de MexicoCDMXMexico
  2. 2.Moscow Institute of Physics and Technology (State University)Dolgoprudnyi, Moscow RegionRussia
  3. 3.Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia
  4. 4.Department of PhysicsTomsk State UniversityTomskRussia

Personalised recommendations