Skip to main content
SpringerLink
Log in

Generalized asymmetric phase-covariant quantum cloning within a nonextensive approach

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, we present a generalized transformation of the optimal asymmetric \(1\longrightarrow 2\) phase-covariant quantum cloning. This generalization is based on the deformed forms of the exponential that emerge from nonextensive statistical mechanics. In particular, two distinct definitions of the q-exponential are discussed. The case where the cloning is symmetric is also studied. In order to highlight the influence of nonextensive treatment on the perfection of clones and entanglement, the effect of the q-index has been clearly illustrated in figures depicting the fidelities in terms of the entanglement parameter \(\theta \) for different values of q. Our study shows that due to the intrinsic properties of the system, the entanglement is not preserved. Thus, entanglement can be controlled by the nonextensive parameter. As an illustration, the incoherent attack on the BB84 protocol has also been considered in the economical case.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Mor, T.: No cloning of orthogonal states in composite systems. Phys. Rev. Lett 80, 3137–3140 (1998)

    Article  ADS  Google Scholar 

  2. Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)

    Article  ADS  Google Scholar 

  3. Dieks, D.: Communication by EPR devices. Phys. Lett. A 92, 271–272 (1982)

    Article  ADS  Google Scholar 

  4. Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145–195 (2002)

    Article  ADS  Google Scholar 

  5. Bužek, V., Hillery, M.: Quantum copying: beyond the no-cloning theorem. Phys. Rev. A 54, 1844–1852 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  6. Scarani, V., Iblisdir, S., Gisin, N., Acin, A.: Quantum cloning. Rev. Mod. Phys. 77, 1225–1256 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Hayashi, M., Iwama, K., Nishimura, H., Raymond, R., Yamashita, S.: Quantum Network Coding. Lecture Notes in Computer Science, vol. 4393. Springer, Berlin (2007)

    MATH  Google Scholar 

  8. Bruß, D., DiVincenzo, D.P., Ekert, A., Fuchs, C.A., Macchiavello, C., Smolin, J.A.: Universal and state-dependent quantum cloning. Phys. Rev. A 57, 2368–2378 (1998)

    Article  ADS  Google Scholar 

  9. Bruß, D., Cinchetti, M., d’ Ariano, G.M., Macchiavello, C.: Phase-covariant quantum cloning. Phys. Rev. A 62, 012302 (2000)

  10. Fuchs, C.A., Gisin, N., Griffiths, R.B., Niu, C.-S., Peres, A.: Optimal eavesdropping in quantum cryptography. I. Information bound and optimal strategy. Phys. Rev. A 56, 1163–1172 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  11. Bruß, D.: Optimal eavesdropping in quantum cryptography with six states. Phys. Rev. Lett. 81, 3018–3021 (1998)

    Article  ADS  Google Scholar 

  12. Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using \(\mathit{d}\)-level systems. Phys. Rev. Lett. 88, 127902 (2002)

    Article  ADS  Google Scholar 

  13. Fan, H., Wang, Y.-N., Jing, L., Yue, J.-D., Shi, H.-D., Zhang, Y.-L., Mu, L.-Z.: Quantum cloning machines and the applications. Phys. Rep. 544(3), 241–322 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  14. Xiong, Z.-X., Shi, H.-D., Wang, Y.-N., Jing, L., Lei, J., Mu, L.-Z., Fan, H.: General quantum key distribution in higher dimension. Phys. Rev. A 85, 012334 (2012)

    Article  ADS  Google Scholar 

  15. Peev, M., Pacher, C., Alléaume, R., Barreiro, C., Bouda, J., Boxleitner, W., Debuisschert, T., Diamanti, E., Dianati, M., Dynes, J., et al.: The secoqc quantum key distribution network in Vienna. N. J. Phys. 11(7), 075001 (2009)

    Article  Google Scholar 

  16. Xu, F., Chen, W., Wang, S., Yin, Z., Zhang, Y., Liu, Y., Zhou, Z., Zhao, Y., Li, H., Liu, D., et al.: Field experiment on a robust hierarchical metropolitan quantum cryptography network. Chin. Sci. Bull. 54(17), 2991–2997 (2009)

    Article  Google Scholar 

  17. Eraerds, P., Walenta, N., Legré, M., Gisin, N., Zbinden, H.: Quantum key distribution and 1 gbps data encryption over a single fibre. N. J. Phys. 12(6), 063027 (2010)

    Article  Google Scholar 

  18. Fröhlich, B., Dynes, J.F., Lucamarini, M., Sharpe, A.W., Yuan, Z., Shields, A.J.: A quantum access network. Nature 501(7465), 69–72 (2013)

    Article  ADS  Google Scholar 

  19. Korzh, B., Lim, C.C.W., Houlmann, R., Gisin, N., Li, M.J., Nolan, D., Sanguinetti, B., Thew, R., Zbinden, H.: Provably secure and practical quantum key distribution over 307 km of optical fibre. Nat. Photon. 9, 163–168 (2015)

    Article  ADS  Google Scholar 

  20. Li, Y.-B., Xu, S.-W., Wang, Q.-L., Liu, F., Wan, Z.-J.: Quantum key distribution based on interferometry and interaction-free measurement. Int. J. Theor. Phys. 1–9 (2015). doi:10.1007/s10773-015-2636-9

  21. Durt, T., Cerf, N.J., Gisin, N., Żukowski, M.: Security of quantum key distribution with entangled qutrits. Phys. Rev. A 67, 012311 (2003)

    Article  ADS  Google Scholar 

  22. Durt, T., Kaszlikowski, D., Chen, J.-L., Kwek, L.C.: Security of quantum key distributions with entangled qudits. Phys. Rev. A 69, 032313 (2004)

    Article  ADS  Google Scholar 

  23. Renner, R., Gisin, N., Kraus, B.: Information-theoretic security proof for quantum-key-distribution protocols. Phys. Rev. A 72, 012332 (2005)

    Article  ADS  Google Scholar 

  24. Chiribella, G., D’Ariano, G.M., Perinotti, P., Cerf, N.J.: Extremal quantum cloning machines. Phys. Rev. A 72, 042336 (2005)

    Article  ADS  Google Scholar 

  25. Tsallis, C.: Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, 1st edn. Springer, New York (2009)

    MATH  Google Scholar 

  26. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Vidiella-Barranco, A.: Entanglement and nonextensive statistics. Phy. Lett. A 260, 335–339 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Tirnakli, U., Özeren, S.F., Büyükkiliç, F., Demirhan, D.: The effect of nonextensivity on the time development of quantum systems. Z. Phys. B Condens. Matter 104(2), 341–345 (1997)

    Article  ADS  Google Scholar 

  29. Naudts, J.: Deformed exponentials and logarithms in generalized thermostatistics. Phys. A 316, 323–334 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tsallis, C.: What are the numbers that experiments provide? Quim. Nova 17, 468–471 (1994)

    Google Scholar 

  31. Cerf, N.J.: Asymmetric quantum cloning in any dimension. Mod. Opt. 47, 187–209 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  32. Durt, T., Du, J.: Characterization of low-cost one-to-two qubit cloning. Phys. Rev. A 69, 062316 (2004)

    Article  ADS  Google Scholar 

  33. Uhlmann, A.: The transition probability in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223(1–2), 1–8 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Fan, H., Matsumoto, K., Wang, X.-B., Wadati, M.: Quantum cloning machines for equatorial qubits. Phys. Rev. A 65, 012304 (2001)

    Article  ADS  Google Scholar 

  37. Rezakhani, A., Siadatnejad, S., Ghaderi, A.: Separability in asymmetric phase-covariant cloning. Phys. Lett. A 336(4–5), 278–289 (2005)

    Article  ADS  MATH  Google Scholar 

  38. Scarani, V., Gisin, N.: Quantum key distribution between N partners: optimal eavesdropping and Bell’s inequalities. Phys. Rev. A 65, 012311 (2001)

    Article  ADS  Google Scholar 

  39. Scarani, V., Iblisdir, S., Gisin, N., Acín, A.: Quantum cloning. Rev. Mod. Phys. 77, 1225–1256 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Theorique, Faculte de Physique, Universite des Sciences et de la Technologie Houari Boumediene, B.P. 32, El-Alia, 16111, Bab-Ezzouar Alger, Algeria

    R. Boudjema, A.-H. Hamici, M. Hachemane & A. Smida

Authors
  1. R. Boudjema
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.-H. Hamici
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Hachemane
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Smida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Boudjema.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boudjema, R., Hamici, AH., Hachemane, M. et al. Generalized asymmetric phase-covariant quantum cloning within a nonextensive approach. Quantum Inf Process 15, 551–563 (2016). https://doi.org/10.1007/s11128-015-1179-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1179-6

Keywords

Search

Navigation