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Time dependence of Eisert–Wilkens–Lewenstein quantum game

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Abstract

The Eisert–Wilkens–Lewenstein (EWL) game can be used to solve the quantum prisoner’s dilemma is investigated. It is assumed that the states of the players are polarized in different directions, and the entangling gate is time dependent, with interaction strength represented by linear, sine, cosine, or exponential functions. If both players cooperate, the payoffs remain above their classical counterparts. However, if they do not cooperate, the payoff for one player increases at the expense of the other. The payoffs of both players are similar when their states are prepared with the same settings, whereas different settings for the initial states result in different payoffs. Due to the periodic nature of the interaction strength, the payoffs oscillate between their classical bounds when both initial states have the same settings. Conversely, for different initial state, the upper bounds are lower than the classical ones, while the minimum values remain above their corresponding classical payoffs.

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The used code of this study is available from the corresponding author upon reasonable request.

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Acknowledgements

We would like to thank the referees for their reports which help us to modify our manuscript. This work was supported in part by the University of Chinese Academy of Sciences.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Women for Art, Science and Education, Ain Shams University, Cairo, Egypt

    A. T. M. Makram-Allah

  2. School of Physics, University of Chinese Academy of Science, Yuquan Road 19A, Beijing, 100049, China

    M. Y. Abd-Rabbou

  3. Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo, 11884, Egypt

    M. Y. Abd-Rabbou

  4. Mathematics Department College of Science, University of Bahrain, PO Box 320038, Sakhir, Bahrain

    N. Metwally

  5. Department of Mathematics, Aswan University, Aswan, Sahari, 81528, Egypt

    N. Metwally

Authors
  1. A. T. M. Makram-Allah
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  2. M. Y. Abd-Rabbou
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  3. N. Metwally
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Contributions

ATM prepared all figures and wrote the original draft. MYA-R performed the mathematical calculations. NM reviewed the draft. All authors read and agreed to the published version of the manuscript.

Corresponding author

Correspondence to M. Y. Abd-Rabbou.

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Appendix

Appendix

In this appendix, we show the elements of the final state are defined by a \(4\times 4 \) matrix, if the states of the player, Alice and Bob, are defined as \(\bigl | \psi _A(0) \bigr \rangle =\bigl | 0 \bigr \rangle _A\), \(\bigl | \psi _B(0) \bigr \rangle =\bigl | 0 \bigr \rangle _B\), respectively. Then the total initial state of the players is given by (6). If players apply their strategies and send their pairs to the entangling gate, then the players follow the steps described in Sect. 2. Finally, the final state is defined by

$$\begin{aligned} \rho _{11}= & \mathbb {C}_{\theta _A}^2 \mathbb {C}_{\theta _B}^2 (\lambda _{\alpha }^-) ^2 \Bigl [\mathbb {C}^2_{2\tilde{\gamma }} (\lambda _{\alpha }^+)^2+\mathbb {S}^2_{2\tilde{\gamma }}\Bigl ] \Bigl [4 \mathbb {C}_{\tilde{\gamma }}^4+4 \mathbb {C}_{\tilde{\gamma }}^2 \Bigl (\mathbb {S}_{\tilde{\gamma }}^2 (\lambda _{\alpha }^+)^2 -1\Bigl )+1\Bigl ], \\ \rho _{22}= & (\lambda _{\alpha }^-)^2 \Bigl [ e^{i \alpha _A} \mathbb {S}_{\theta _B}A \mathbb {C}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }}( \lambda _{B})\mathbb {C}_{2\tilde{\gamma }}-i \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i (2 \alpha _A+\alpha _B)} \mathbb {C}_{2\tilde{\gamma }} ^2 \\ & +i \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i \alpha _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ]\times \Bigl [ e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }}(\lambda _{B}) \mathbb {C}_{2\tilde{\gamma }} +i \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i \alpha _B} \mathbb {C}_{2\tilde{\gamma }} ^2\\ & -i \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i (2 \alpha _A+\alpha _B)} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ], \\ \rho _{33}= & ie^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \lambda _{A} \mathbb {C}_{2\tilde{\gamma }} -e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \Bigl [-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _B} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl ], \\ \rho _{44}= & -(\lambda _{\alpha }^-) ^2 \Bigl [ \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B}\mathbb {S}_{2\tilde{\gamma }} \mathbb {C}_{2\tilde{\gamma }} \Bigl ((\lambda _{\alpha }^+) ^2 -1\Bigl )-i \mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B}(\lambda _{\alpha }^+)\Bigl ] ^2, \\ \rho _{12}= & \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B}(\lambda _{\alpha }^-) ^2 \Bigl [(\lambda _{\alpha }^+) ^2 \mathbb {C}^2_{2\tilde{\gamma }}+\mathbb {S}^2_{2\tilde{\gamma }}\Bigl ] \\ & \times \Bigl [ i e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }}(\lambda _{B}) \mathbb {C}_{2\tilde{\gamma }}- e^{i \alpha _B} \mathbb {S}_{\theta _B} \mathbb {C}_{\theta _A}\Bigl (-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _A} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl )\Bigl ],\\ \rho _{13}= & \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B}(\lambda _{\alpha }^-) ^2 \Bigl [(\lambda _{\alpha }^+) ^2 \mathbb {C}^2_{2\tilde{\gamma }}+\mathbb {S}^2_{2\tilde{\gamma }}\Bigl ]0 \\ & \times \Bigl [ ie^{i \alpha _B} \mathbb {S}_{\theta _B} \mathbb {C}_{\theta _A} \mathbb {S}_{2\tilde{\gamma }}\lambda _{A}\mathbb {C}_{2\tilde{\gamma }} -e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \Bigl (-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _B} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl )\Bigl ], \\ \rho _{14}= & \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B}(\lambda _{\alpha }^-) ^2 \Bigl [(\lambda _{\alpha }^+) ^2 \mathbb {C}^2_{2\tilde{\gamma }}+\mathbb {S}^2_{2\tilde{\gamma }}\Bigl ]\times \Bigl [ i\mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }} ((\lambda _{\alpha }^+) ^2-1) \mathbb {C}_{2\tilde{\gamma }}+\mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B} (\lambda _{\alpha }^+)\Bigl ], \\ \rho _{21}= & -\mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B}(\lambda _{\alpha }^-) ^2 \Bigl [4 \mathbb {C}_{\tilde{\gamma }}^4+4 \mathbb {C}_{\tilde{\gamma }}^2 \Bigl ((\lambda _{\alpha }^+) ^2 \mathbb {S}_{\tilde{\gamma }}^2-1\Bigl )+1\Bigl ] \\ & \times \Bigl [ i e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} (\lambda _{B}) \mathbb {C}_{2\tilde{\gamma }} +\mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i (2 \alpha _A+\alpha _B)}\mathbb {C}^2_{2\tilde{\gamma }} - e^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ], \\ \rho _{23}= & (\lambda _{\alpha }^-) ^2 \Bigl [ e^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }}\lambda _{A} \mathbb {C}_{2\tilde{\gamma }} +i e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \Bigl (-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _B} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl )\Bigl ] \\ & \times \Bigl [ e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B}\mathbb {S}_{2\tilde{\gamma }}(\lambda _{B}) \mathbb {C}_{2\tilde{\gamma }} -i \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i (2 \alpha _A+\alpha _B)}\mathbb {C}^2_{2\tilde{\gamma }} + ie^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ], \end{aligned}$$
$$\begin{aligned} \rho _{24}= & (\lambda _{\alpha }^-) ^2 \Bigl [2 e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}_{\tilde{\gamma }}\mathbb {C}_{\tilde{\gamma }} (\lambda _{B}) \mathbb {C}_{2\tilde{\gamma }} -i \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} e^{i (2 \alpha _A+\alpha _B)}\mathbb {C}^2_{2\tilde{\gamma }}\\ & \times + i e^{i \alpha _B}\mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ]0 \Bigl [\mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }} \mathbb {C}_{2\tilde{\gamma }}\Bigl ((\lambda _{\alpha }^+) ^2-1\Bigl ) -i \mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B} (\lambda _{\alpha }^+)\Bigl ], \\ \rho _{31}= & (\lambda _{\alpha }^-) ^2 \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \Bigl [4 \mathbb {C}_{\tilde{\gamma }}^4+4 \mathbb {C}_{\tilde{\gamma }}^2 \Bigl ((\lambda _{\alpha }^+) ^2 \mathbb {S}_{\tilde{\gamma }}^2-1\Bigl )+1\Bigl ] \\ & \times \Bigl [- ie^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \lambda _{A}\mathbb {C}_{2\tilde{\gamma }} -\mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} e^{i (\alpha _A+2 \alpha _B)}\mathbb {C}^2_{2\tilde{\gamma }} + e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ], \\ \rho _{32}= & (\lambda _{\alpha }^-) ^2 \Bigl [ i e^{i \alpha _B}\mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \lambda _{A}\mathbb {C}_{2\tilde{\gamma }} +\mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} e^{i (\alpha _A+2 \alpha _B)}\mathbb {C}^2_{2\tilde{\gamma }} - e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ] \\ & \times \Bigl [ e^{i \alpha _B}\mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \Bigl (-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _A} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl )- i e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} (\lambda _{B}) \mathbb {C}_{2\tilde{\gamma }} \Bigl ], \\ \rho _{34}= & (\lambda _{\alpha }^-) ^2 \Bigl [ e^{i \alpha _B}\mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B}\mathbb {S}_{2\tilde{\gamma }} \lambda _{A} \mathbb {C}_{2\tilde{\gamma }} -i \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} e^{i (\alpha _A+2 \alpha _B)}\mathbb {C}^2_{2\tilde{\gamma }} + i e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}^2_{2\tilde{\gamma }}\Bigl ] \\ & \times \Bigl [ \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {C}_{2\tilde{\gamma }} \Bigl ((\lambda _{\alpha }^+) ^2-1\Bigl )-i \mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B} e^{i (\alpha _A+\alpha _B)}\Bigl ], \\ \rho _{41}= & \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B}(\lambda _{\alpha }^-) ^2 \Bigl [4 \mathbb {C}_{\tilde{\gamma }}^4+4 \mathbb {C}_{\tilde{\gamma }}^2 \Bigl ((\lambda _{\alpha }^+) ^2 \mathbb {S}_{\tilde{\gamma }}^2-1\Bigl )+1\Bigl ]\\ & \times \Bigl [ i \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }} \mathbb {C}_{2\tilde{\gamma }}\Bigl ((\lambda _{\alpha }^+) ^2-1\Bigl )+\mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B} (\lambda _{\alpha }^+)\Bigl ],\\ \rho _{42}= & (\lambda _{\alpha }^-) ^2 \Bigl [- e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} (\lambda _{B})\mathbb {C}_{2\tilde{\gamma }} -i e^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \Bigl (-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _A} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl )\Bigl ] \\ & \times \Bigl [2 \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {C}_{\tilde{\gamma }}S_{\gamma } \mathbb {C}_{2\tilde{\gamma }}\Bigl ((\lambda _{\alpha }^+) ^2-1\Bigl )-i \mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B} (\lambda _{\alpha }^+)\Bigl ], \\ \rho _{43}= & (\lambda _{\alpha }^-) ^2 \Bigl [ ie^{i \alpha _B} \mathbb {C}_{\theta _A} \mathbb {S}_{\theta _B} \mathbb {S}_{2\tilde{\gamma }} \lambda _{A}\mathbb {C}_{2\tilde{\gamma }} -e^{i \alpha _A} \mathbb {S}_{\theta _A} \mathbb {C}_{\theta _B} \Bigl (-4 \mathbb {C}_{\tilde{\gamma }}^2 (1+e^{2 i \alpha _B} \mathbb {S}_{\tilde{\gamma }}^2)+4 \mathbb {C}_{\tilde{\gamma }}^4+1\Bigl )\Bigl ] \\ & \times \Bigl [i \mathbb {C}_{\theta _A} \mathbb {C}_{\theta _B} \mathbb {C}_{2\tilde{\gamma }}\Bigl ((\lambda _{\alpha }^+) ^2-1\Bigl ) +\mathbb {S}_{\theta _A} \mathbb {S}_{\theta _B} (\lambda _{\alpha }^+)\Bigl ]. \end{aligned}$$

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Makram-Allah, A.T.M., Abd-Rabbou, M.Y. & Metwally, N. Time dependence of Eisert–Wilkens–Lewenstein quantum game. Quantum Inf Process 23, 393 (2024). https://doi.org/10.1007/s11128-024-04589-2

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