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Quantum Information Processing

, Volume 14, Issue 1, pp 147–164 | Cite as

Variable entangling in a quantum prisoner’s dilemma cellular automaton

  • Ramón Alonso-Sanz
Article

Abstract

The effect of variable entangling on the dynamics of a spatial quantum formulation of the iterated prisoner’s dilemma game is studied in this work. The game is played in the cellular automata manner, i.e., with local and synchronous interaction. The effect of spatial structure is assessed when allowing the players to adopt quantum and classical strategies, both in the two- and three-parameter strategy spaces.

Keywords

Quantum games Entangling Spatial Cellular automata 

Notes

Acknowledgments

This work was supported by the Spanish Grant M2012-39101-C02-01. Part of the computations of this work were performed in EOLO and FISWULF, HPC machines of the International Campus of Excellence of Moncloa, funded by the Spanish Government and Feder Funds.

References

  1. 1.
    Alonso-Sanz, R.: A quantum prisoner’s dilemma cellular automaton. Proc. R. Soc. A 470, 20130793 (2014)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alonso-Sanz, R.: On a three-parameter quantum battle of the sexes cellular automaton. Quantum Inf. Process. 12(5), 1835–1850 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alonso-Sanz, R.: A quantum battle of the sexes cellular automaton. Proc. R. Soc. A 468, 3370–3383 (2012)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alonso-Sanz, R.: Dynamical Systems with Memory. World Scientific Pub., Singapore (2011)Google Scholar
  5. 5.
    Benjamin, S.C., Hayden, P.M.: Comment on “quantum games and quantum strategies”. Phys. Rev. Lett. 87(6), 069801 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    Binmore, K.: Fun and Games. D.C.Heath, Lexington (1992)zbMATHGoogle Scholar
  7. 7.
    Bleiler, S. : A Formalism for Quantum Games and an Application. http://arxiv.org/abs/0808.1389 (2008)
  8. 8.
    Branderburger, A.: The relationship between quantum and classical correlation games. Games Econ. Behav. 89, 157–183 (2010)Google Scholar
  9. 9.
    Du, J.F., Xu, X.D., Li, H., Zhou, X., Han, R., et al.: Entanglement playing a dominating role in quantum games. Phys. Lett. A 289(1–2), 9–15 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Du, J.F., Li, H., Xu, X.D., Zhou, X., Han, R.: Phase-transition-like behaviour of quantum games. J. Phys. A Math. Gen. 36(23), 6551–6562 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Du, J.F., Xu, X.D., Li, H., Zhou, X., Han, R.: Entanglement enhanced multiplayer quantum games. Phys. Lett. A 302, 222–233 (2002)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83(15), 3077–3080 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Eisert, J., Wilkens, M.: Quantum games. J. Mod. Opt. 47(14–15), 2543–2556 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Flitney, A.P., Hollenberg, L.C.L.: Nash equilibria in quantum games with generalized two-parameter strategies. Phys. Lett. A 363, 381–388 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Flitney, A.P., Abbott, D.: Advantage of a quantum player over a classical one in 2x2 quantum games. Proc. R. Soc. Lond. A 459(2038), 2463–2474 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Flitney A.P., Abbott, D.: An introduction to quantum game theory. Fluct. Noise Lett. 02, R175. http://arxiv.org/pdf/quant-ph/0208069 (2002)
  17. 17.
    Khan, F.S., Phoenix, S.J.D.: Gaming the quantum. Quantum Inf. Comput. 13(3–4), 231–244. http://arxiv.org/pdf/1202.1142 (2013)
  18. 18.
    Landsburg, S.E.: Quantum game theory. In: The Wiley Encyclopedia of Operations Research and Management Science. http://arxiv.org/pdf/1110.6237v1 (2011)
  19. 19.
    Landsburg, S.E.: Quantum game theory. Not. AMS. http://www.ams.org/notices/200404/fea-landsburg (2004)
  20. 20.
    Levine, D.K. : Quantum Games Have No News for Economists. http://levine.sscnet.ucla.edu/papers/quantumnonews (2005)
  21. 21.
    Li, Q., Iqbal, A., Perc, M., Chen, M., Abbott, D.: Coevolution of quantum and classical strategies on evolving random networks. PloS One 8(7), e68423 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    Li, Q., Iqbal, A., Chen, M., Abbott, D.: Quantum strategies win in a defector-dominated population. Physica A 391, 3316–3322 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    Li, Q., Iqbal, A., Chen, M., Abbott, D.: Evolution of quantum and classical strategies on networks by group interactions. New J. Phys. 14(10), 103034 (2012)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272, 291–303 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052–1055 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Miszczak, J.A., Pawela, L., Sladkowski, J. : General model for an entanglement-enhanced composed quantum game on a two-dimensional lattice. Fluct. Noise Lett. 13(2), 1450012. http://arxiv.org/abs/1306.4506 (2014)
  27. 27.
    Nawaz, A., Toor, A.H.: Dilemma and quantum battle of sexes. J. Phys. A Math. Gen. 37(15), 4437–4443 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Nawaz, A., Toor, A.H.: Generalized quantization scheme for two-person non-zero sum games. J. Phys. A Math. Gen. 37(42), 365305 (2004)Google Scholar
  29. 29.
    Nowak, N.M., May, R.M.: Evolutionary games and spatial chaos. Nature 359, 826–829 (1992)ADSCrossRefGoogle Scholar
  30. 30.
    Nowak, M.A., May, R.M.: The spatial dilemmas of evolution. Int. J. Bifurc. Chaos 3(11), 35–78 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Owen, G.: Game Theory. Academic Press, Waltham (1995)zbMATHGoogle Scholar
  32. 32.
    Phoenix, S.J.D., Khan, F.S.: The role of correlations in classical and quantum games. Fluct. Noise Lett. 12(3), 1350011 (2013)CrossRefGoogle Scholar
  33. 33.
    Piotrowski, E.W., Sladkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42(5), 1089–1099 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Schiff, J.L.: Cellular Automata: A Discrete View of the World. Wiley, New York (2008)Google Scholar
  35. 35.
    Wiesner, K.: Quantum Cellular automata. Encycl. Complex. Syst. Sci., 7154–7164. http://arxiv.org/abs/0808.0679 (2009)

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.ETSIA (Estadística, GSC)Technical University of MadridMadridSpain

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