Effect of partial-collapse measurement on quantum Stackelberg duopoly game in noninertial frame

  • Xiang-Ping LiaoEmail author
  • Chang-Ning Pan
  • Man-Sheng Rong
  • Mao-Fa Fang


An efficient method is proposed to improve the payoffs of the firms in quantum Stackelberg duopoly game under noninertial frame using partial-collapse measurement. It is shown that the payoffs of firms can be enhanced to a great extent. We obtain the maximally retrievable payoffs of firms for maximally entangled initial state and unentangled initial state respectively. In particular, the payoffs of firms approach to certain constants and Nash equilibrium exists for the whole range values of the acceleration parameter by changing partial-collapse measurement strength. Our work provides a novel method to recover payoffs in quantum game under Unruh decoherence and exhibits the ability of partial-collapse measurement as an important technique in relativistic quantum game.


Stackelberg duopoly game Noninertial frame The payoff Partial-collapse measurement 



This work was supported by the National Natural Science Foundation of China (Grant Nos. 11374096 and 11604094), the Natural Science Foundation of Hunan Province of China (Grant No. 2016JJ2044) and the Major Program for the Research Foundation of Education Bureau of Hunan Province of China (Grant No. 16A057).


  1. 1.
    Straffin, P.D.: Game Theory and Strategy. The Mathematical Association of America, Washington (1993)zbMATHGoogle Scholar
  2. 2.
    Binmore, K.: Playing for Real: A Text on Game Theory. Oxford University Press, New York (2007)CrossRefGoogle Scholar
  3. 3.
    Samuelson, P.A., Nordhaus, W.D.: Economics. McGraw-Hill, New York (1993)Google Scholar
  4. 4.
    Li, H., Du, J., Massar, S.: Continuous-variable quantum games. Phys. Lett. A 306, 73 (2002)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Du, J., Li, H., Ju, C.: Quantum games of asymmetric information. Phys. Rev. E 68, 016124 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    Lo, C.F., Kiang, D.: Quantum Bertrand duopoly with differentiated products. Phys. Lett. A 321, 94 (2004)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bierman, H.S., Fernandez, L.: Game Theory with Economic Applications, 2nd edn. Addison Wesley, Reading (1998)Google Scholar
  8. 8.
    Lo, C.F., Kiang, D.: Quantum Stackelberg duopoly. Phys. Lett. A 318, 333 (2003)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bierman, H.S., Fernandez, L.: Game Theory with Economic Applications. Addison Wesley, Reading (1998)Google Scholar
  10. 10.
    Gravelle, H., Rees, R.: Microeconomics. Longman Harlow, New York (1992)Google Scholar
  11. 11.
    Rasmusen, E.: Games and Information: An Introduction to Game Theory. Peking University, Beijing (2003). (in Chinese)zbMATHGoogle Scholar
  12. 12.
    Zhang, W.Y.: Game Theory and Information Economics. Shanghai People’s Press, Shanghai (2004). (in Chinese)Google Scholar
  13. 13.
    Alsing, P.M., Fuentes, I.: Focus issue on relativistic quantum information. Class. Quantum Grav. 29, 224001 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Situ, H.Z., Huang, Z.M.: Relativistic quantum Bayesian game under decoherence. Int. J. Theor. Phys. 55, 2354 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Khan, S., Khan, M.K.: Noisy relativistic quantum games in noninertial frames. Quantum Inf. Process. 12, 1351 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    Goudarzi, H., Beyrami, S.: Effect of uniform acceleration on multiplayer quantum game. J. Phys. A Math. Theor. 45, 225301 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    Khan, S., Khan, M.K.: Relativistic quantum games in noninertial frames. J. Phys. A Math. Theor. 44, 355302 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    Khan, S., Khan, M.K.: Quantum Stackelberg duopoly in a noninertial frame. Chin. Phys. Lett. 28, 070202 (2011)CrossRefGoogle Scholar
  19. 19.
    Katz, N., et al.: Coherent state evolution in a superconducting qubit from partial-collapse measurement. Science 312, 1498 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    Blok, M.S., Bonato, C., Markham, M.L., Twitchen, D.J., Dobrovitski, V.V., Hanson, R.: Manipulating a qubit through the backaction of sequential partial measurements and real-time feedback. Nat. Phys. 10, 189 (2014)CrossRefGoogle Scholar
  21. 21.
    Sun, Q.Q., Al-Amri, M., Zubairy, M.S.: Reversing the weak measurement of an arbitrary field with finite photon number. Phys. Rev. A 80, 033838 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Man, Z.X., Xia, Y.J., An, N.B.: Manipulating entanglement of two qubits in a common environment by means of weak measurements and quantum measurement reversals. Phys. Rev. A 86, 012325 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    Liao, X.P., Ding, X.Z., Fang, M.F.: Improving the payoffs of cooperators in three-player cooperative game using weak measurements. Quantum Inf. Process. 14, 4395 (2015)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Xiao, X., Xie, Y.M., Yao, Y., Li, Y.L., Wang, J.C.: Retrieving the lost fermionic entanglement by partial measurement in noninertial frames. Ann. Phys. 390, 83 (2018)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kim, Y.S., Cho, Y.W., Ra, Y.S., Kim, Y.H.: Reversing the weak quantum measurement for a photonic qubit. Opt. Express 17, 11978 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    Kim, Y.S., Lee, J.C., Kwon, O., Kim, Y.H.: Protecting entanglement from decoherence using weak measurement and quantum measurement reversal. Nat. Phys. 8, 117 (2012)CrossRefGoogle Scholar
  27. 27.
    Xiao, X., Yao, X., Zhong, W.J., Li, Y.L., Xie, Y.M.: Enhancing teleportation of quantum Fisher information by partial measurements. Phys. Rev. A 93, 012307 (2016)ADSCrossRefGoogle Scholar
  28. 28.
    Gibbons, R.: Game Theory for Applied Economists. Princeton University Press, Princeton (1992)Google Scholar
  29. 29.
    Khan, S., Ramzan, M., Khan, M.K.: Quantum model of Bertrand duopoly. Chinese Phys. Lett. 27, 080302 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    Lo, C.F., Kiang, D.: Quantum Stackelberg duopoly with incomplete information. Phys. Lett. A 346, 65 (2005)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Wang, X., Hu, C.Z.: Quantum Stackelberg duopoly with continuous distributed incomplete information. Chin. Phys. Lett. 29, 120303 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272, 291 (2000)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Iqbal, A., Toor, A.H.: Backwards-induction outcome in a quantum game. Phys. Rev. A 65, 052328 (2002)ADSCrossRefGoogle Scholar
  34. 34.
    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)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhu, X., Kuang, L.M.: Quantum Stackelberg duopoly game in depolarizing channel. Commun. Theor. Phys. 49, 111 (2008)ADSMathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  37. 37.
    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)ADSCrossRefGoogle Scholar
  38. 38.
    Crispino, L.C.B., Higuchi, A., Matsas, G.E.A.: The Unruh effect and its applications. Rev. Mod. Phys. 80, 787 (2008)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Wang, S.C., Yu, Z.W., Zou, W.J., Wang, X.B.: Protecting quantum states from decoherence of finite temperature using weak measurement. Phys. Rev. A 89, 022318 (2014)ADSCrossRefGoogle Scholar
  40. 40.
    Liao, X.P., Fang, M.F., Fang, J.S., Zhu, Q.Q.: Preserving entanglement and the fidelity of three-qubit quantum states undergoing decoherence using weak measurement. Chin. Phys. B 23, 020304 (2014)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Xiang-Ping Liao
    • 1
    Email author
  • Chang-Ning Pan
    • 1
  • Man-Sheng Rong
    • 1
  • Mao-Fa Fang
    • 2
  1. 1.College of ScienceHunan University of TechnologyZhuzhouChina
  2. 2.School of Physics and ElectronicsHunan Normal UniversityChangshaChina

Personalised recommendations