International Journal of Theoretical Physics

, Volume 56, Issue 3, pp 863–873 | Cite as

Quantum Samaritan’s Dilemma Under Decoherence

  • Zhiming Huang
  • Ramón Alonso-Sanz
  • Haozhen Situ


We study how quantum noise affects the solution of quantum Samaritan’s dilemma. Serval most common dissipative and nondissipative noise channels are considered as the model of the decoherence process. We find that the solution of quantum Samaritan’s dilemma is stable under the influence of the amplitude damping, the bit flip and the bit-phase flip channel.


Quantum game Noise channel Nash equilibrium 



We are very grateful to the reviewers and the editors for their invaluable comments and detailed suggestions that helped to improve the quality of the present paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 61502179, 61472452), the Natural Science Foundation of Guangdong Province of China (Grant No. 2014A030310265), Guangdong Province Office of Education (Grant No. 2014KTSCX130), the Science Foundation for Young Teachers of Wuyi University (Grant No. 2015zk01), the Spanish Grant MTM2015-63914-P. H.Z. Situ is sponsored by the State Scholarship Fund of the China Scholarship Council.


  1. 1.
    Myerson, R.B.: Game theory: Analysis of conflict. Havard University Press, Boston (1991)zbMATHGoogle Scholar
  2. 2.
    Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eisert, J., Wilkens, M.: Quantum games. J. Mod. Opt. 47, 2543 (2000)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Benjamin, S.C., Hayden, P.M.: Comment on “Quantum Games and Quantum Strategies”. Phys. Rev. Lett. 87, 069801 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    Benjamin, S.C., Hayden, P.M.: Multiplayer quantum games. Phys. Rev. A 64, 030301 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    Khan, S., Khan, M.K.: Relativistic quantum games in noninertial frames. J. Phys. A - Math. Theor. 44, 355302 (2011)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Situ, H.Z.: A quantum approach to play asymmetric coordination games. Quant. Inf. Process. 13, 591 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fra̧ckiewicz, P.: A new quantum scheme for normal-form games. Quant. Inf. Process. 14, 1809 (2015)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fra̧ckiewicz, P.: Remarks on quantum duopoly schemes. Quant. Inf. Process. 15, 121 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Weng, G.F., Yu, Y.: Playing quantum games by a scheme with pre- and post-selection. Quant. Inf. Process. 15, 147 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khan, S., Khan, M.K.: Quantum Stackelberg duopoly in a noninertial frame. Chinese Phys. Lett. 28, 070202 (2011)CrossRefGoogle Scholar
  12. 12.
    Fra̧ckiewicz, P.: Quantum signaling game. J. Phys. A - Math. Theor. 47, 305301 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fra̧ckiewicz, P., Sładkowski, J.: Quantum information approach to the ultimatum game. Int. J. Theor. Phys. 53, 3248 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fra̧ckiewicz, P.: On signaling games with quantum chance move. J. Phys. A - Math. Theor. 48, 075303 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Anand, N., Benjamin, C.: Do quantum strategies always win?. Quant. Inf. Process. 14, 4027 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iqbal, A., Abbott, D.: Quantum matching pennies game. J. Phys. Soc. Jpn. 78, 014803 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    Brunner, N., Linden, N.: Connection between Bell nonlocality and Bayesian game theory. Nat. Commun. 4, 2057 (2013)ADSGoogle Scholar
  18. 18.
    Iqbal, A., Chappell, J.M., Li, Q., Pearce, C.E.M., Abbott, D.: A probabilistic approach to quantum Bayesian games of incomplete information. Quant. Inf. Process. 13, 2783 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pappa, A., Kumar, N., Lawson, T., Santha, M., Zhang, S.Y., Diamanti, E., Kerenidis, I.: Nonlocality and conflicting interest games. Phys. Rev. Lett. 114, 020401 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Iqbal, A., Chappell, J.M., Abbott, D.: Social optimality in quantum Bayesian games. Phys. A 436, 798 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Situ, H.Z.: Quantum Bayesian game with symmetric and asymmetric information. Quant. Inf. Process. 14, 1827 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Situ, H.Z.: Two-player conflicting interest Bayesian games and Bell nonlocality. Quant. Inf. Process. 15, 137 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Situ, H.Z., Huang, Z.M.: Relativistic quantum Bayesian game under decoherence. Int. J. Theor. Phys. 55, 2354 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Khan, S., Ramzan, M., Khan, M.K.: Decoherence effects on multiplayer cooperative quantum games. Commun. Theor. Phys. 56, 228 (2011)CrossRefzbMATHGoogle Scholar
  25. 25.
    Fra̧ckiewicz, P.: N-person quantum Russian roulette. Phys. A 401, 8 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Liao, X.P., Ding, X.Z., Fang, M.F.: Improving the payoffs of cooperators in three-player cooperative game using weak measurements. Quant. Inf. Process. 14, 4395 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Situ, H.Z., Zhang, C., Yu, F.: Quantum advice enhances social optimality in three-party conflicting interest games. Quantum Inf. Comput. 16, 588 (2016)MathSciNetGoogle Scholar
  28. 28.
    Alonso-Sanz, R.: A quantum battle of the sexes cellular automaton. P. Roy. Soc. A - Math. Phy. 468, 3370 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Alonso-Sanz, R.: On a three-parameter quantum battle of the sexes cellular automaton. Quant. Inf. Process. 12, 1835 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Alonso-Sanz, R.: A quantum prisoner’s dilemma cellular automaton. P. Roy. Soc. A - Math. Phy. 470, 20130793 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Alonso-Sanz, R.: Variable entangling in a quantum prisoner’s dilemma cellular automaton. Quant. Inf. Process. 14, 147 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Alonso-Sanz, R.: A cellular automaton implementation of a quantum battle of the sexes game with imperfect information. Quant. Inf. Process. 14, 3639 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ozdemir, S.K., Shimamura, J., Morikoshi, F., Imoto, N.: Dynamics of a discoordination game with classical and quantum correlations. Phys. Lett. A 333, 218 (2004)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Flitney, A.P., Abbott, D.: Quantum games with decoherence. J. Phys. A: Math. Gen. 38, 449 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nawaz, A., Toor, A.H.: Quantum games with correlated noise. J. Phys. A: Math. Gen. 39, 9321 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Khan, S., Ramzan, M., Khan, M.K.: Quantum Stackelberg duopoly in the presence of correlated noise. J. Phys. A: Math. Theor. 43, 375301 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Khan, S., Khan, M.K.: Noisy relativistic quantum games in noninertial frames. Quant. Inf. Process. 12, 1351 (2013)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Ramzan, M.: Three-player quantum Kolkata restaurant problem under decoherence. Quant. Inf. Process 12, 577 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ramzan, M., Khan, M.K.: Environment-assisted quantum minority games. Fluct. Noise Lett. 12, 1350025 (2013)CrossRefGoogle Scholar
  40. 40.
    Gawron, P., Kurzyk, D., Pawela, L.: Decoherence effects in the quantum qubit flip game using Markovian approximation. Quant. Inf. Process. 13, 665 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Huang, Z.M., Qiu, D.W.: Quantum games under decoherence. Int. J. Theor. Phys. 55, 965 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Economics and ManagementWuyi UniversityJiangmenChina
  2. 2.Technical University of Madrid, ETSIA (Estadística, GSC). C.UniversitariaMadridSpain
  3. 3.College of Mathematics and InformaticsSouth China Agricultural UniversityGuangzhouChina

Personalised recommendations