Skip to main content
SpringerLink
Log in

Improving the payoffs of cooperators in three-player cooperative game using weak measurements

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, an efficient method is proposed to improve the payoffs of cooperators in cooperative three-player quantum game under the action of amplitude damping, bit flip and depolarizing channels using weak measurements. It is shown that the payoffs of cooperators can be enhanced to a great extent in the case of amplitude damping channel, and the payoff sudden death can be avoided in the case of bit flip and depolarizing channels. Moreover, the payoffs of cooperators tend to a constant by changing weak measurement strength in spite of sufficiently strong decoherence.

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

Similar content being viewed by others

References

  1. von-Neumann, J., Morgenstein, O.: The Theory of Games and Economic Behaviour. Princeton University Press, Princeton, NJ (1944)

    Google Scholar 

  2. Nash, J.: Equilibrium points in n-person games. Proc. Nat. Acad. Sci. 36, 48 (1950)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Cheon, T., Iqbal, A.: Bayesian Nash equilibria and Bell inequalities. J. Phys. Soc. Jpn. 77, 024801 (2008)

    Article  ADS  Google Scholar 

  8. Iqbal, A., Abbott, D.: Quantum matching pennies game. J. Phy. Soc. Jpn. 78, 014803 (2009)

    Article  ADS  Google Scholar 

  9. Iqbal, A., Cheon, T., Abbott, D.: Probabilistic analysis of three-player symmetric quantum games played using the Einstein–Podolsky–Rosen–Bohm setting. Phys. Lett. A 372, 6564 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Iqbal, A., Toor, A.H.: Evolutionarily stable strategies in quantum games. Phys. Lett. A 280, 249 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. Iqbal, A., Toor, A.H.: Quantum mechanics gives stability to Nash equilibrium. Phys. Rev. A 65, 022036 (2002)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. 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  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Johnson, N.F.: Playing a quantum game with a corrupted source. Phys. Rev. A 63, 020302(R) (2001)

    Article  ADS  MathSciNet  Google Scholar 

  18. DAriano, G.M., Gill, R.D., Keyl, M., Kuemmerer, B., Maassen, H., Werner, R.F.: The quantum Monty Hall problem. Quant. Inf. Comp. 2, 355 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Miszczak, J.A., Gawron, P., Puchala, Z.: Qubit flip game on a Heisenberg spin chain. arXiv:1108.0642 [quant-ph], (2011)

  20. Puya, S., Hoshang, H.: Quantum solution to a three player Kolkata restaurant problem using entangled qutrits. arXiv:1111.1962 [quant-ph], (2011)

  21. Chakrabarti, A.S., Chakrabarti, B.K., Chatterjee, A., Mitra, M.: The Kolkata paise restaurant problem and resource utilization. Phys. A 388, 2420–2426 (2009)

    Article  Google Scholar 

  22. Benjamin, S.C., Hayden, P.M.: Multiplayer quantum games. Phys. Rev. A 64, 030301(R) (2001)

    Article  ADS  Google Scholar 

  23. Chen, Q., Wang, Y.: N-player quantum minority game. Phys. Lett. A A 327, 98,102 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Flitney, A.P., Greentree, A.D.: Coalitions in the quantum Minority game: classical cheats and quantum bullies. Phys. Lett. A 362, 132 (2007)

    Article  ADS  MATH  Google Scholar 

  25. Schmid, C., Flitney, A.P.: Experimental implementation of a four-player quantum game. arXiv:0901.0063v1 [quant-ph], (2008)

  26. Flitney, A.P., Hollenberg, L.C.L.: Multiplayer quantum minority game with decoherence. Quantum Inf. Comput. 7, 111 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Khan, S., Ramzan, M., Khan, M.K.: Quantum Parrondos games under decoherence. Int. J. Theor. Phys 49, 31 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khan, S., Ramzan, M., Khan, M.K.: Quantum Monty Hall problem under decoherence. Commun. Theor. Phys. 54, 47 (2010)

    Article  ADS  MATH  Google Scholar 

  30. Gawron, P., Miszczak, J.A., Sladkowski, J.: Noise effects in quantum magic squares game. Int. J. Quant. Inf. 6, 667 (2008)

    Article  MATH  Google Scholar 

  31. Gawron, P.: Noisy quantum Monty Hall game. Fluct. Noise Lett. 9, 9 (2010)

    Article  MathSciNet  Google Scholar 

  32. 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 

  33. 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  MathSciNet  ADS  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  MathSciNet  ADS  MATH  Google Scholar 

  35. Ramzan, M.: Three-player quantum Kolkata restaurant problem under decoherence. Quantum Inf. Process. 12(1), 577–586 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Khan, S., Ramzan, M., Khan, M.K.: Decoherence effects on multiplayer cooperative quantum games. Commun. Theor. Phys. 56, 228–234 (2011)

    Article  MATH  Google Scholar 

  37. Gawron, P., Kurzyk, D., Pawela, L.: Decoherence effects in the quantum qubit flip game using Markovian approximation. Quantum. Inf. Process. 13, 665–682 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)

    Article  ADS  Google Scholar 

  39. Aharonov, Y., Botero, A., Pospescu, S., Reznik, B., Tollaksen, J.: Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values. Phys. Lett. A 301, 130 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Lundeen, J.S., Steinberg, A.M.: Experimental joint weak measurement on a photon pair as a probe of Hardys paradox. Phys. Rev. Lett. 102, 020404 (2009)

    Article  ADS  Google Scholar 

  41. Yokota, K., Yamamoto, T., Koashi, M., Imoto, N.: Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair. N. J. Phys. 11, 033011 (2009)

    Article  Google Scholar 

  42. Korotkov, A.N., Jordan, A.N.: Undoing a weak quantum measurement of a solid-state qubit. Phys. Rev. Lett. 97, 166805 (2006)

    Article  ADS  Google Scholar 

  43. 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)

    Article  ADS  Google Scholar 

  44. Lee, J.C., Jeong, Y.C., Kim, Y.S., Kim, Y.H.: Experimental demonstration of decoherence suppression via quantum measurement reversal. Opt. Express 19, 16309 (2011)

    Article  ADS  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. Man, Z.X., Xia, Y.J., An, N.B.: Enhancing entanglement of two qubits undergoing independent decoherences by local pre- and postmeasurements. Phys. Rev. A 86, 052322 (2012)

    Article  ADS  Google Scholar 

  47. Burger, E., Freund, J.E.: Introduction to the Theory of Games. Prentice-Hall, Englewood Cliffs (1963)

    Google Scholar 

  48. 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)

    Article  ADS  Google Scholar 

  49. Du, J., Li, H., Xu, X., Shi, M., Wu, J., Zhou, X., Han, R.: Experimental realization of quantum games on a quantum computer. Phys. Rev. Lett. 88, 137902 (2002)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  51. Kolenderski, P., Sinha, U., Youning, L., Zhao, T., Volpini, M., Cabello, A., Laflamme, R., Jennewein, T.: Aharon–Vaidman quantum game with a Young-type photonic qutrit. Phys. Rev. A 86, 012321 (2012)

    Article  ADS  Google Scholar 

  52. Korotkov, A.N., Jordan, A.N.: Undoing a weak quantum measurement of a solid-state qubit. Phys. Rev. Lett. 97, 166805 (2006)

    Article  ADS  Google Scholar 

  53. Katz, N., Neeley, M., Ansmann, M., Bialczak, R.C., Hofheinz, M., Lucero, E., OConnell, A., Wang, H., Cleland, A.N., Martinis, J.M., Korotkov, A.N.: Reversal of the weak measurement of a quantum state in a superconducting phase qubit. Phys. Rev. Lett. 101, 200401 (2008)

    Article  ADS  Google Scholar 

  54. 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)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No.11374096) and the Major Program for the Research Foundation of Education Bureau of Hunan Province of China (Grant No. 10A026).

Author information

Authors and Affiliations

  1. College of Science, Hunan University of Technology, Zhuzhou, 412008, Hunan, China

    Xiang-Ping Liao

  2. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410081, Hunan, China

    Xiang-Zhuo Ding

  3. College of Physics and Information Science, Hunan Normal University, Changsha, 410081, Hunan, China

    Mao-Fa Fang

Authors
  1. Xiang-Ping Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang-Zhuo Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mao-Fa Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiang-Ping Liao.

Appendices

Appendix 1

In this Appendix, we present the payoff function of cooperators A and B by using Eq. (5) for the amplitude damping channel using weak measurements as follows:

$$\begin{aligned} P_{A,B}^{AD2}= & {} \left[ n^6\, q\, {\cos \left( \frac{Q}{2}\right) }^2 + n^6\, r\, {\cos \left( \frac{Q}{2}\right) }^2 + m^6\, q\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \right. \nonumber \\&-\, 2\, n^6\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 -3\, m^6\, p\, q\, {\sin \left( \frac{Q}{2}\right) }^2 - 3\, m^6\, p\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&-\, 2\, m^6\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, p^2\, q\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, 3\, m^6\, p^2\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 6\, m^6\, p\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&-\, m^6\, n^2\, p\, q\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, n^2\, p\, r\, {\sin \left( \frac{Q}{2}\right) }^2- 6\, m^6\, p^2\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, 2\, m^6\, p^3\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 2\, m^6\, n^2\, p^2\, q\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, n^2\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&-\, m^6\, n^4\, p^2\, q\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, n^4\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, n^6\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, 2\, m^6\, n^2\, p^2\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, n^2\, p^3\, r\,{\sin \left( \frac{Q}{2}\right) }^2- m^6\, n^4\, p^2\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, m^6\, n^4\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, n^6\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - 4\, m^6\, n^2\, p^2\, q\, r\,{\sin \left( \frac{Q}{2}\right) }^2 \,\nonumber \\&+\, 2\, m^6\, n^2\, p^3\, q\, r {\sin \left( \frac{Q}{2}\right) }^2 + 2\, m^6\, n^4\, p^2\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \,\nonumber \\&-\, 2\, m^6\, n^4\, p^3\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - 2\, m^6\, n^6\, p^3\, q\, r\,{\sin \left( \frac{Q}{2}\right) }^2\nonumber \\&+\, 2\, m^6\, n^2\, p\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 +4\, m^3\, n^3\, \sqrt{q}\, \sqrt{r}\, \cos \left( \frac{Q}{2}\right) \, \sin \left( \frac{Q}{2}\right) \, {\left( 1 - p\right) }^{\frac{3}{2}}\,\nonumber \\&\sqrt{1 - q}\, \sqrt{1 - r}[/]n^6\, {\cos \left( \frac{Q}{2}\right) }^2+ m^6\, {\sin \left( \frac{Q}{2}\right) }^2 - 3\, m^6\, p\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, 3\, m^6\, p^2\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2- 6\, m^6\, n^2\, p^2\,{\sin \left( \frac{Q}{2}\right) }^2\nonumber \\&+\,3\, m^6\, n^2\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, n^4\, p^2\, {\sin \left( \frac{Q}{2}\right) }^2 - 3\, m^6\, n^4\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, m^6\, n^6\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2+3\, m^6\, n^2\, p\, {\sin \left( \frac{Q}{2}\right) }^2 + 8\, m^3\, n^3\, \sqrt{q}\, \sqrt{r}\, \cos \left( \frac{Q}{2}\right) \,\nonumber \\&\left. \sin \left( \frac{Q}{2}\right) \, {\left( 1 - p\right) }^{\frac{3}{2}}\sqrt{1 - q}\, \sqrt{1 - r}\right] \end{aligned}$$

When the initial state is maximally entangled, i.e., \(Q=\pi /2\), we obtain the payoff function Eq. (10).

Appendix 2

In this Appendix, we present the payoff function of cooperators A and B by using Eq. (5) for the bit flip channel using weak measurements as follows:

$$\begin{aligned} P_{A,B}^{Bf2}= & {} \left[ n^6\, q\, {\cos \left( \frac{Q}{2}\right) }^2 + n^6\, r\, {\cos \left( \frac{Q}{2}\right) }^2 + p^3\, q\, {\cos \left( \frac{Q}{2}\right) }^2 + p^3\, r\, {\cos \left( \frac{Q}{2}\right) }^2 \right. \\&+\, m^6\, q\, {\sin \left( \frac{Q}{2}\right) }^2 +7 m^6\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - n^4\, p\, q\, {\cos \left( \frac{Q}{2}\right) }^2 \\&-\, 3\, n^6\, p\, q\, {\cos \left( \frac{Q}{2}\right) }^2 - n^4\, p\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - 3\, n^6\, p\, r\, {\cos \left( \frac{Q}{2}\right) }^2 \\&-\, 2\, n^6\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - 2\, p^3\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - 3\, m^6\, p\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \\&-\, 3\, m^6\, p\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - 2\, m^6\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2- n^2\, p^2\, q\, {\cos \left( \frac{Q}{2}\right) }^2 \\&+\, n^2\, p^3\, q\, {\cos \left( \frac{Q}{2}\right) }^2 + 2\, n^4\, p^2\, q\, {\cos \left( \frac{Q}{2}\right) }^2 - n^4\, p^3\, q\, {\cos \left( \frac{Q}{2}\right) }^2 \\&+\, 3\, n^6\, p^2\, q\, {\cos \left( \frac{Q}{2}\right) }^2- n^6\, p^3\, q\, {\cos \left( \frac{Q}{2}\right) }^2 - n^2\, p^2\, r\, {\cos \left( \frac{Q}{2}\right) }^2 \\&+\, n^2\, p^3\, r\, {\cos \left( \frac{Q}{2}\right) }^2 + 2\, n^4\, p^2\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - n^4\, p^3\, r\, {\cos \left( \frac{Q}{2}\right) }^2 \\&+\, 3\, n^6\, p^2\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - n^6\, p^3\, r\, {\cos \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, p^2\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \\&-\, m^6\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, p^2\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \\ \end{aligned}$$
$$\begin{aligned}&+\, 6\, m^6\, p\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 2\, n^2\, p^2\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - 2\, n^2\, p^3\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 \\&-\, 4\, n^4\, p^2\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 + 2\, n^4\, p^3\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - 6\, n^6\, p^2\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 \\&+\, 2\, n^6\, p^3\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - m^6\, n^2\, p\, q\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, n^2\, p\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \\&-\, 6\, m^6\, p^2\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 2\, m^6\, p^3\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 2\, m^6\, n^2\, p^2\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \\&-\, m^6\, n^2\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, n^4\, p^2\, q\, {\sin \left( \frac{Q}{2}\right) }^2+ m^6\, n^4\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 \\&+\, m^6\, n^6\, p^3\, q\, {\sin \left( \frac{Q}{2}\right) }^2 + 2\, m^6\, n^2\, p^2\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, n^2\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2\\&-\, m^6\, n^4\, p^2\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, n^4\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, n^6\, p^3\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \\&+\, 2\, n^4\, p\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2+ 6\, n^6\, p\, q\, r\, {\cos \left( \frac{Q}{2}\right) }^2 - 4\, m^6\, n^2\, p^2\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \\&+\, 2\, m^6\, n^2\, p^3\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 2\, m^6\, n^4\, p^2\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 \\&-\, 2\, m^6\, n^4\, p^3\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 - 2\, m^6\, n^6\, p^3\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2\\ \end{aligned}$$
$$\begin{aligned}&+\, 2\, m^6\, n^2\, p\, q\, r\, {\sin \left( \frac{Q}{2}\right) }^2 + 4\, m^3\, n^3\, \sqrt{q}\, \sqrt{r}\, \cos \left( \frac{Q}{2}\right) \,\\&\sin \left( \frac{Q}{2}\right) \, \sqrt{1 - q}\, \sqrt{1 - r}- 16\, m^3\, n^3\, p\, \sqrt{q}\, \sqrt{r}\, \cos \left( \frac{Q}{2}\right) \,\\&\sin \left( \frac{Q}{2}\right) \, \sqrt{1 - q}\, \sqrt{1 - r} + 16\, m^3\, n^3\, p^2\, \sqrt{q}\, \sqrt{r}\,\\&\cos \left( \frac{Q}{2}\right) \, \sin \left( \frac{Q}{2}\right) \, \sqrt{1 - q}\, \sqrt{1 - r}[/]n^6\, {\cos \left( \frac{Q}{2}\right) }^2+ p^3\, {\cos \left( \frac{Q}{2}\right) }^2\\&+ m^6\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, n^4\, p\, {\cos \left( \frac{Q}{2}\right) }^2 - 3\, n^6\, p\, {\cos \left( \frac{Q}{2}\right) }^2 - 3\, m^6\, p\, {\sin \left( \frac{Q}{2}\right) }^2\\&+\, 3\, n^2\, p^2\, {\cos \left( \frac{Q}{2}\right) }^2 - 3\, n^2\, p^3\, {\cos \left( \frac{Q}{2}\right) }^2 - 6\, n^4\, p^2\, {\cos \left( \frac{Q}{2}\right) }^2 + 3\, n^4\, p^3\, {\cos \left( \frac{Q}{2}\right) }^2 \nonumber \\&+\, 3\, n^6\, p^2\, {\cos \left( \frac{Q}{2}\right) }^2 - n^6\, p^3\, {\cos \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, p^2\, {\sin \left( \frac{Q}{2}\right) }^2 - m^6\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&-\, 6\, m^6\, n^2\, p^2\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, n^2\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, n^4\, p^2\, {\sin \left( \frac{Q}{2}\right) }^2\nonumber \\&-\, 3\, m^6\, n^4\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2 + m^6\, n^6\, p^3\, {\sin \left( \frac{Q}{2}\right) }^2 + 3\, m^6\, n^2\, p\, {\sin \left( \frac{Q}{2}\right) }^2 \nonumber \\&\left. +\, 8\, m^3\, n^3\, \sqrt{q}\, \sqrt{r}\, \cos \left( \frac{Q}{2}\right) \, \sin \left( \frac{Q}{2}\right) \, \sqrt{1 - q}\, \sqrt{1 - r}\right] \end{aligned}$$

When the initial state is maximally entangled, i.e., \(Q=\pi /2\), we obtain the payoff function Eq. (16).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, XP., Ding, XZ. & Fang, MF. Improving the payoffs of cooperators in three-player cooperative game using weak measurements. Quantum Inf Process 14, 4395–4412 (2015). https://doi.org/10.1007/s11128-015-1144-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1144-4

Keywords

Search

Navigation