Skip to main content
SpringerLink
Log in

Quantum Games under Decoherence

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Quantum systems are easily influenced by ambient environments. Decoherence is generated by system interaction with external environment. In this paper, we analyse the effects of decoherence on quantum games with Eisert-Wilkens-Lewenstein (EWL) (Eisert et al., Phys. Rev. Lett. 83(15), 3077 1999) and Marinatto-Weber (MW) (Marinatto and Weber, Phys. Lett. A 272, 291 2000) schemes. Firstly, referring to the analytical approach that was introduced by Eisert et al. (Phys. Rev. Lett. 83(15), 3077 1999), we analyse the effects of decoherence on quantum Chicken game by considering different traditional noisy channels. We investigate the Nash equilibria and changes of payoff in specific two-parameter strategy set for maximally entangled initial states. We find that the Nash equilibria are different in different noisy channels. Since Unruh effect produces a decoherence-like effect and can be perceived as a quantum noise channel (Omkar et al., arXiv:1408.1477v1), with the same two parameter strategy set, we investigate the influences of decoherence generated by the Unruh effect on three-player quantum Prisoners’ Dilemma, the non-zero sum symmetric multiplayer quantum game both for unentangled and entangled initial states. We discuss the effect of the acceleration of noninertial frames on the the game’s properties such as payoffs, symmetry, Nash equilibrium, Pareto optimal, dominant strategy, etc. Finally, we study the decoherent influences of correlated noise and Unruh effect on quantum Stackelberg duopoly for entangled and unentangled initial states with the depolarizing channel. Our investigations show that under the influence of correlated depolarizing channel and acceleration in noninertial frame, some critical points exist for an unentangled initial state at which firms get equal payoffs and the game becomes a follower advantage game. It is shown that the game is always a leader advantage game for a maximally entangled initial state and there appear some points at which the payoffs become zero.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Myerson, R.B.: Game Theory: Analysis of Conflict. MIT Press, Cambridge (1991)

    MATH  Google Scholar 

  2. Meyer, D.A.: Quantum strategies. Phys. Rev. Lett 82, 1052 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and guantum strategies. Phy. Rev. Lett 83(15), 3077 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272, 291 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Flitney, A.P., Abbott, D.: Quantum version of the Monty Hall problem. Phys. Rev. A 65, 062318 (2002)

    Article  ADS  Google Scholar 

  6. Eisert, J., Wilkens, M.: Quantum games. J. Mod. Opt 47, 2543 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Flitney, A.P., Ng, J., Abbott, D: Quantum Parrondo’s games. Physica A 314, 35 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Iqbal, A., Toor, A.H.: Quantum cooperative games. Phys. Lett. A 293, 103 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Situ, H.Z.: A quantum approach to play asymmetric coordination games. Quant. Inf. Process 13, 591–599 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Situ, H.Z.: Quantum Bayesian game with symmetric and asymmetric information. Quant. Inf. Process. 14, 1827–1840 (2015)

  11. Flitney, A.P., Abbott, D.: Advantage of a quantum player over a classical one in 2 ×. quantum games. Proc. R. Soc. Lond. A 459, 2463–2474 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Ichikawa, T., Tsutsui, I.: Duality, phase structures, and dilemmas in symmetric quantum games. Ann. Phys. 322, 531 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Ichikawa, T., Tsutsui, I., Cheon, T.J.: Quantum game theory based on the Schmidt decomposition. J. Phys. A: Math. Theor. 41, 135303 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Khan, S., Ramzan, M., Khan, M.K.: Quantum Model of Bertrand Duopoly. Chin. Phys. Lett. 27, 080302 (2010)

    Article  ADS  Google Scholar 

  15. Flitney, A.P., Abbott, D.: Quantum games with decoherence. J. Phys. A 38, 449 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Chen, L.K., Ang, H., Kiang, D., Kwek, L.C., Lo, C.F.: Quantum prisoner dilemma under decoherence. Phys. Lett. A 316, 317 (2003)

    Article  ADS  MATH  Google Scholar 

  17. Chen, J.L., Kwek, L.C., Oh, C.H.: Noisy quantum game. Phys. Rev. A 65, 052320 (2002)

    Article  ADS  Google Scholar 

  18. Khan, S., Ramzan, M., Khan, M.K.: Quantum Parrondo’s games under decoherence. Int. J. Theor. Phys 49, 31–41 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ramzan, M.: Three-player quantum Kolkata restaurant problem under decoherence. Quantum Inf. Process 12, 577–586 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Zhu, X., Kuang, L.M.: Quantum Stackelberg duopoly game in depolarizing channel. Commun. Theor. Phys 49, 111 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  21. Zhu, X., Kuang, L.M.: The influence of entanglement and decoherence on the quantum Stackelberg duopoly game. J. Phys. A Math. Theor 40, 7729 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Omkar, S., Subhashish, B., Srikanth, R., Ashutosh, K.A.: The Unruh effect interpreted as a quantum noise channel. arXiv:1408.1477v1

  23. Ling, Y., He, S., Qiu, W., Zhang, H.: Quantum entanglement of electromagnetic field in non-inertial reference frames. J. Phys. A Math. Theor 40, 9025–9032 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Pan, Q., Jing, J.: Degradation of nonmaximal entanglement of scalar and dirac fields in noninertial frames. Phys. Rev. A 77, 024302 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  25. Khan, S., Khan, M.K.: Open quantum systems in noninertial frames. J. Phys. A Math. Theor. 44, 045305 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Wang, J., Jing, J.: Quantum decoherence in noninertial frames. Phys. Rev. A 82, 032324 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Wang, J., Jing, J., Fan, H.: Quantum discord and measurement-induced disturbance in the background of dilaton black holes. Phys. Rev. D 90, 025032 (2014)

  28. Khan, S., Khan, M.K.: Quantum Stackelberg Duopoly in a noninertial frame. Chin. Phys. Lett. 28, 070202 (2011)

    Article  Google Scholar 

  29. Khan, S., Khan, M.K.: Relativistic quantum games in noninertial frames. J. Phys. A: Math. Theor. 44, 355302 (2011)

    Article  ADS  MATH  Google Scholar 

  30. Goudarzi, H., Beyrami, S.: Effect of uniform acceleration on multiplayer quantum game. J. Phys. A: Math. Theor. 45, 225301 (2012)

    Article  ADS  MATH  Google Scholar 

  31. Khan, S., Khan, M.K.: Noisy relativistic quantum games in noninertial frames. Quantum Inf. Process 12, 1351–1363 (2013)

    Article  ADS  MATH  Google Scholar 

  32. Ramzan, M., Khan, M.K.: Noise effects in a three-player prisoner’s dilemma quantum game. J. Phys. A Math. Theor. 41, 435302 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Nawaz, A., Toor, A.H.: Quantum games with correlated noise. J. Phys. A: Math. Gen 39, 9321 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Ramzan, M., Nawaz, A., Toor, A.H., Khan, M.K.: The effect of quantum memory on quantum games. J. Phys. A Math. Theor. 41, 055307 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Khan, S., Ramzan, M., Khan, M.K.: Quantum Stackelberg duopoly in the presence of correlated noise. J. Phys. A: Math. Theor. 43, 375301 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Benjamin, S.C., Hayden, P.M.: Comment on “Quantum games and quantum strategies”. Phys. Rev. Lett. 87, 069801 (2001)

    Article  ADS  Google Scholar 

  37. Eisert, J., Wilkens, M., Lewenstein, M.: Comment on “Quantum games and quantum strategies”-reply. Phy. Rev. Lett. 87, 069802 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  38. Flitney, A.P., Hollenberg, L.C.L.: Nash equilibria in quantum games with generalized two-parameter strategies. Phys. Lett. A 363, 381 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Prevedel, R., Stefanov, A., Walther, P., Zeilinger, A.: Experimental realization of a quantum game on a one-way quantum computer. New J. Phys. 9, 205 (2007)

    Article  ADS  Google Scholar 

  40. Nielson, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  41. Du, J., Li, H., Xu, X., Zhou, X., Han, R.: Entanglement enhanced multiplayer quantum games. Phys. Lett. A 302, 229–233 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Alsing, P.M., Fuentes-Schuller, I., Mann, R.B., Tessier, T.E.: Entanglement of Dirac fields in noninertial frames. Phys. Rev. A 74, 032326 (2006)

    Article  ADS  Google Scholar 

  43. Davies, P.C.W.: Scalar production in Schwarzschild and Rindler metrics. J. Phys. A: Math. Gen 8, 609 (1975)

    Article  ADS  Google Scholar 

  44. Takagi, S.: Vacuum noise and stress induced by uniform acceleration. Hawking-Unruh effect in Rindler manifold of arbitrary dimension. Prog. Theor. Phys. Suppl 88, 1 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  45. Gibbons, R.: Game Theory for Applied Economists. Princeton University Press, Princeton (1992)

    Google Scholar 

Download references

Acknowledgments

We are thankful to the anonymous referees for valuable suggestions to improve the quality of the paper. This work is supported in part by the National Natural Science Foundation of China (Nos. 61272058, 61073054), the Natural Science Foundation of Guangdong Province of China (No. 10251027501000004), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20100171110042).

Author information

Authors and Affiliations

  1. Department of Computer Science, Sun Yat-sen University, Guangzhou, 510006, China

    Zhiming Huang & Daowen Qiu

  2. The Guangdong Key Laboratory of Information Security Technology, Sun Yat-sen University, Guangzhou, 510006, China

    Zhiming Huang & Daowen Qiu

  3. School of Economics and Management, Wuyi University, Jiangmen, 529020, China

    Zhiming Huang

Authors
  1. Zhiming Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daowen Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daowen Qiu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Z., Qiu, D. Quantum Games under Decoherence. Int J Theor Phys 55, 965–992 (2016). https://doi.org/10.1007/s10773-015-2741-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-015-2741-9

Keywords

Search

Navigation