Advertisement

Entanglement and coherence in quantum prisoner’s dilemma

  • Alan C. SantosEmail author
Article
  • 46 Downloads

Abstract

Entanglement and coherence are quantum resources widely used in several tasks in quantum information processing. In particular, the emergence of quantum game theory arises the question whether entanglement would be more useful than coherence for quantum players. In this paper, we address such question from a two-person quantum game, namely the quantum version of the prisoner’s dilemma. We discuss the players payoffs (i) when an entangled initial game state is provided and (ii) when the system is started in a separable superposition state. As the main result, when an entangled state is provided to players, we find a situation where a non-maximally entangled state is preferable by a quantum player concerning the maximally entangled state. Thus, our first result suggests that we can establish a trade-off between maximum expected payoff and an amount of entanglement required by a quantum player. As a second result, when we provide a non-entangled initial state (but we have coherence), the payoff of a classical player is enhanced concerning the previous case. We discuss how the phase-transition-like behavior emerges from entanglement in the game considered here, so that we could design a game where no change in optimal strategies would be required.

Keywords

Quantum game Prisoner dilemma Coherence Entanglement 

Notes

Acknowledgements

This work is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil). The author also acknowledges the financial support in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES) (Finance Code 001) and by the Brazilian National Institute for Science and Technology of Quantum Information [CNPq INCT-IQ (465469/2014-0)]

References

  1. 1.
    Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052 (1999)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077 (1999)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Maioli, A., Passos, M., Balthazar, W., Souza, C., Huguenin, J., Schmidt, A.: Quantization and experimental realization of the colonel blotto game. Quantum Inf. Process. 18(1), 10 (2019)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Anand, N., Benjamin, C.: Do quantum strategies always win? Quantum Inf. Process. 14(11), 4027 (2015)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Balakrishnan, S., Sankaranarayanan, R.: Classical rules and quantum strategies in penny flip game. Quantum Inf. Process. 12(2), 1261 (2013)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Du, J., Xu, X., Li, H., Zhou, X., Han, R.: Playing prisoner’s dilemma with quantum rules. Fluct. Noise Lett. 2(04), R189 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Du, J., Li, H., Xu, X., Zhou, X., Han, R.: Entanglement enhanced multiplayer quantum games. Phys. Lett. A 302(5–6), 229 (2002)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Benjamin, S.C., Hayden, P.M.: Multiplayer quantum games. Phys. Rev. A 64, 030301 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    Shi, L., Xu, F.: Nonlinear dynamics of a quantum cournot duopoly game with heterogeneous players. Quantum Inf. Process. 18(7), 227 (2019)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Mendes, R.V.: The quantum ultimatum game. Quantum Inf. Process. 4(1), 1 (2005)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ramzan, M., Khan, M.: Distinguishing quantum channels via magic squares game. Quantum Inf. Process. 9(6), 667 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khan, F.S., Solmeyer, N., Balu, R., Humble, T.S.: Quantum games: a review of the history, current state, and interpretation. Quantum Inf. Process. 17(11), 309 (2018)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    de Ponte, M.A., Santos, A.C.: Adiabatic quantum games and phase-transition-like behavior between optimal strategies. Quantum Inf. Process. 17(6), 149 (2018)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Du, J., Li, H., Xu, X., Zhou, X., Han, R.: Phase-transition-like behaviour of quantum games. J. Phys. A: Math. Theor. 36(23), 6551 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. Cambridge University Press, New York (2011)zbMATHGoogle Scholar
  16. 16.
    Audenaert, K., Plenio, M.B., Eisert, J.: Entanglement cost under positive-partial-transpose-preserving operations. Phys. Rev. Lett. 90, 027901 (2003)ADSCrossRefGoogle Scholar
  17. 17.
    Plenio, M.B.: Logarithmic negativity: a full entanglement monotone that is not convex. Phys. Rev. Lett. 95, 090503 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Gour, G., Marvian, I., Spekkens, R.W.: Measuring the quality of a quantum reference frame: the relative entropy of frameness. Phys. Rev. A 80, 012307 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)ADSCrossRefGoogle Scholar
  22. 22.
    Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Shao, L.H., Xi, Z., Fan, H., Li, Y.: Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A 91, 042120 (2015)ADSCrossRefGoogle Scholar
  24. 24.
    Yuan, X., Zhou, H., Cao, Z., Ma, X.: Intrinsic randomness as a measure of quantum coherence. Phys. Rev. A 92, 022124 (2015)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Instituto de FísicaUniversidade Federal FluminenseNiteróiBrazil

Personalised recommendations