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On the effect of memory in a quantum prisoner’s dilemma cellular automaton

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Abstract

The disrupting effect of quantum memory on the dynamics of a spatial quantum formulation of the iterated prisoner’s dilemma game with variable entangling is studied. The game is played within a cellular automata framework, i.e., with local and synchronous interactions. The main findings of this work refer to the shrinking effect of memory on the disruption induced by noise.

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Notes

  1. \(\gamma ^\star _{0.0,0.0}\!=\!\arcsin \left( \sqrt{\frac{P-S}{\mathfrak {T}-S}=\frac{1}{4}}\right) =\frac{\pi }{6} \simeq 0.524, \gamma ^\bullet _{0.0,0.0}\!=\!\arcsin \left( \sqrt{\frac{\mathfrak {T}-R}{\mathfrak {T}-S}=\frac{2}{4}}\right) =\frac{\pi }{4}\!\simeq 0.785\).

  2. \(\gamma ^\star _{0.5,0.0}\!=\!2\arcsin \left( \sqrt{\frac{P-S}{2(\mathfrak {T}-S)}\!=\!\frac{1}{8}}\right) \!\simeq \!0.723, \gamma ^\bullet _{0.5,0.0}\!=\!2\arcsin \left( \sqrt{\frac{\mathfrak {T}-R}{2(\mathfrak {T}-S)}\!=\!\frac{2}{8}}\right) \!=\! \frac{\pi }{3} \simeq 1.047\).

  3. From Eq. (6), it is \(p_B^{ \mathcal Q,\mathcal D(u)}(0)=\mathfrak {T}\cos ^2\gamma + S\sin ^2\gamma \) and \(p_{A,B}^{\mathcal Q,\mathcal C(u)}(0)=R\cos ^2\gamma + P\sin ^2\gamma \), whose equalization leads to \(\gamma ^\bullet _{0.0,0.0}=\arcsin \left( \sqrt{\frac{\mathfrak {T}-R}{\mathfrak {T}+P-R-S}}\right) = \arcsin \left( \sqrt{\frac{2}{3}}\right) \simeq 0.955\). In this scenario, it is \(\gamma ^\bullet _{0.5,0.0}=2\arcsin \left( \sqrt{\frac{P-S}{\mathfrak {T}+P-R-S}=\frac{1}{3}}\right) \simeq 1.231\).

  4. It is \(\Pi ^{\hat{\mathcal M},\hat{U}(\theta _B, 0)}_{u}(0)\!=\!\displaystyle \frac{1}{2} \left( \begin{matrix} 0 &{} 0\\ 1-\sin \theta _B &{} 1+\sin \theta _B \end{matrix}\right) \), which renders \(p^{\hat{\mathcal M},\hat{U}(\theta _B, 0) }_{\left\{ \!\begin{array}{c}\!A\!\\ \!B \!\end{array}\!\right\} } (0, 0) =\displaystyle \frac{1}{2}\left[ \left\{ \!\begin{array}{c}\!\mathfrak {T}\\ \!S\end{array}\!\right\} (1-\sin \theta _B) +P(1+\sin \theta _B)\right] \). Consequently, \(p^{\hat{\mathcal M},\hat{U}(\theta _B, 0)}_{\left\{ \!\begin{array}{c}\!A\!\\ \!B \!\end{array}\!\right\} }(0,0)=P\) for \(\theta _B=\pi /2.\)

  5. \(p^{\hat{\mathcal W},\hat{U}(\pi /2, 0)}_A(0,0)=\frac{1}{4}\left[ 11+\frac{3}{4}(2+\sqrt{2})\sin ^2\gamma -3\sqrt{2+\sqrt{2}}\sin \gamma \right] \), and \(p^{\hat{\mathcal W},\hat{U}(\pi /2, 0)}_B(0,0)=\frac{1}{4}\left[ 11-\frac{5}{4}(2+\sqrt{2})\sin ^2\gamma + \sqrt{2+\sqrt{2}}\sin \gamma \right] \). These payoffs are structurally similar to those generated by the \(\{\hat{\mathcal M},\hat{U}(\pi /2, 0)\}\) pair from \(\Pi ^{\hat{\mathcal M},\hat{U}(\pi /2, 0)} _u (0,0) =\frac{1}{4} \left( \begin{array}{cc}\cos ^2\gamma &{}\cos ^2\gamma \\ (\cos \frac{\gamma }{2}-\sin \frac{\gamma }{2})^4 &{} (\cos \frac{\gamma }{2}+\sin \frac{\gamma }{2})^4 \end{array}\right) \), which are equal to \(p^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}_A(0,0)\!=\!\frac{1}{4}(11+ 3\sin ^2\gamma - 6\sin \gamma )\), and \(p^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}_B(0,0)\!=\!\frac{1}{4}(11 - 5\sin ^2\gamma + 2\sin \gamma )\). In both games it is \(p_A^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}(0,0)=p_B^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}(0,0)=11/4\) at \(\gamma =0\), and \(p_B^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}(0,0)>p_A^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}(0,0)\), even for \(\gamma =\pi /2\) in the \(\{\hat{\mathcal W},\hat{U}(\pi /2, 0)\}\) game, whereas \(p_A^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}(0,0)=p_B^{\hat{\mathcal M},\hat{U}(\pi /2, 0)}(0,0)=2\) in the \(\{\hat{\mathcal M},\hat{U}(\pi /2, 0)\}\) game.

  6. It is \( \Pi ^{\mathcal M,\mathcal D}_u (0,0) = \frac{1}{2} \left( \begin{array}{cc} 0&{} \sin ^2\gamma \\ \cos ^2\gamma &{} 1\end{array}\right) \), which renders the payoffs \(p_{_{\texttt {A}}}^{\mathcal M,\mathcal D}(0, 0)= \frac{1}{2}\left[ (\mathfrak {T}-S)\sin ^2\gamma + P+S\right] ,\) and \(p_{_{\texttt {B}}}^{\mathcal M,\mathcal D}(0,0)= \frac{1}{2}\left[ (S-\mathfrak {T})\sin ^2\gamma + P+\mathfrak {T}\right] .\)

  7. The values of \((p_A,p_B)\) at \(\gamma =\pi /2\) in the \((\hat{\mathcal M},\hat{\mathcal D})\) two-player game, and those of \((\overline{p}_A,\overline{p}_B)\) in the CA simulations of Fig. 3 are (from top to bottom): (3.125, 2.125), (3.042, 2.181); (3.203, 1.996), (3.110, 2.071); (3.280, 1.866), (3.166, 1.944).

References

  1. Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83(15), 3077–3080 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Alonso-Sanz, R.: On a three-parameter quantum battle of the sexes cellular automaton. Quantum Inf. Process. 12(5), 1835–1850 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Alonso-Sanz, R.: A quantum battle of the sexes cellular automaton. Proc. R. Soc. A 468, 3370–3383 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Huang, Z., Alonso-Sanz, R., Situ, H.: Quantum Samaritan’s dilemma under decoherence. Int. J. Theor. Phys. 56(3), 863–873 (2017)

    Article  MATH  Google Scholar 

  5. Alonso-Sanz, R., Situ, H.: A quantum Samaritan’s dilemma cellular automaton. R. Soc. Open Sci. 4(6), 863–160669 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ozdemir, S.K., Shimamura, J., Morikoshi, F., Imoto, N.: Dynamics of a discoordination game with classical and quantum correlations. Phys. Lett. A 333, 218–231 (2004)

    Article  ADS  MATH  Google Scholar 

  7. Du, J.F., Li, H., Xu, X., Shi, M., Wi, J., Zhou, X., Han, R.: Experimental realization of quantum games on a quantum computer. Phys. Rev. Lett. 88(13), 137902 (2002)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  9. 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 

  10. Owen, G.: Game Theory. Academic Press, Cambridge (1995)

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  14. Eisert, J., Wilkens, M.: Quantum games. J. Mod. Opt. 47(14–15), 2543–2556 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Alonso-Sanz, R.: Spatial correlated games. R. Soc. Open Sci. 4(6), 171361 (2017). https://doi.org/10.1098/rsos.171361

    Article  Google Scholar 

  16. Zhang, S. (2012). Quantum strategic game theory. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference (ITCS ’12). ACM, New York. ISBN: 978-1-4503-1115-1

  17. Brunner, N., Linden, N.: Connection between nonlocality and Bayesian game theory. Nat. Commun. 4, 2057 (2013)

    ADS  Google Scholar 

  18. Pappa, A., Kumar, N., Lawson, T., Santha, M., Zhang, S., Diamanti, E., Kerenidis, I.: Nonlocality and conflicting interest games. Phys. Rev. Lett. 114, 020401 (2015)

    Article  ADS  Google Scholar 

  19. Yin, J., et al.: Satellite-based entanglement distribution over 1200 kilometers. Science 356, 1140 (2017)

    Article  Google Scholar 

  20. Schiff, J.L.: Cellular Automata: A Discrete View of the World. Wiley, London (2008)

    MATH  Google Scholar 

  21. Du, J.F., Xu, X.D., Li, H., Zhou, X., Han, R.: Entanglement playing a dominating role in quantum games. Phys. Lett. A 89(1–2), 9–15 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Alonso-Sanz, R.: On the effect of quantum noise in a quantum prisoner’s dilemma cellular automaton. Quantum Inf. Process. 16(6), 161 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Alonso-Sanz, R.: Variable entangling in a quantum prisoner’s dilemma cellular automaton. Quantum Inf. Process. 14(1), 147–164 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Alonso-Sanz, R.: A quantum prisoner’s dilemma cellular automaton. Proc. R. Soc. A 470, 20130793 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Flitney, A.P., Abbott, D.: Advantage of a quantum player over a classical one in \(2\times 2\) quantum games. Proc. R. Soc. Lond. A 459(2038), 2463–2474 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Flitney, A.P., Abbott, D.: Quantum games with decoherence. J. Phys. A Math. Gen. 38(2), 449 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Alonso-Sanz, R., Situ, H.: On the effect of quantum noise in a quantum relativistic prisoner’s dilemma cellular automaton. Int. J. Theor. Phys. 55(12), 5265–5279 (2016)

    Article  MATH  Google Scholar 

  29. Alonso-Sanz, R., Situ, H.: On the effect of quantum noise in a quantum relativistic battle of the sexes cellular automaton. Phys. A 468, 267–277 (2016)

    Article  MATH  Google Scholar 

  30. Alonso-Sanz, R.: Dynamical Systems with Memory. World Scientific, Singapore (2011)

    Book  MATH  Google Scholar 

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Acknowledgements

This work has been funded by the Spanish Grant MTM2015-63914-P. Part of the computations of this work were performed in EOLO, an HPC machine of the International Campus of Excellence of Moncloa, funded by the UCM and Feder Funds.

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Authors and Affiliations

  1. Grupo de Sistemas Complejos, ETSIAAB, Universidad Politécnica de Madrid, Avda. Puerta de Hierro 2-4, 28040, Madrid, Spain

    Ramón Alonso-Sanz & Fabio Revuelta

  2. Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049, Madrid, Spain

    Fabio Revuelta

Authors
  1. Ramón Alonso-Sanz
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Correspondence to Ramón Alonso-Sanz.

Appendix

Appendix

The thresholds for a fair quantum game played with the (5, 3, 2, 1)-PD parameters and a noise strength equal to \( \mu = 0.5\) are given as a function of the memory factor \(\eta \) by

$$\begin{aligned} \gamma _{0.5,\eta }^*&= 2\arcsin \left( \frac{2}{\sqrt{18+(10\sqrt{2} - 9) \eta + \sqrt{196 + (280 \sqrt{2} -244) \eta + ( 281 - 180 \sqrt{2} ) \eta ^2}}}\right) , \\ \gamma _{0.5,\eta }^\bullet&= 2\arcsin \left( \frac{2 \sqrt{2}}{\sqrt{20+(10\sqrt{2} - 11) \eta + \sqrt{144 + 40 (6 \sqrt{2} - 7) \eta + ( 321 - 220 \sqrt{2} ) \eta ^2 }}}\right) . \end{aligned}$$
(A.1a)

For the unfair quantum versus classical case, the lowest threshold is given by Eq. (A.1a), while the largest one equals

$$\begin{aligned} \gamma _{0.5,\eta }^\bullet = 2\arcsin \left( \frac{2 \sqrt{2}}{\sqrt{16+(6\sqrt{2} - 7) \eta + \sqrt{64 + (96 \sqrt{2} - 128) \eta + ( 121 - 84 \sqrt{2} ) \eta ^2 }}}\right) . \end{aligned}$$
(A.1b)

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Alonso-Sanz, R., Revuelta, F. On the effect of memory in a quantum prisoner’s dilemma cellular automaton. Quantum Inf Process 17, 60 (2018). https://doi.org/10.1007/s11128-018-1823-z

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