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.
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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).
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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:
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:
When the initial state is maximally entangled, i.e., \(Q=\pi /2\), we obtain the payoff function Eq. (16).
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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
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DOI: https://doi.org/10.1007/s11128-015-1144-4