Advertisement

Quantum Information Processing

, Volume 11, Issue 6, pp 1571–1583 | Cite as

Qubit flip game on a Heisenberg spin chain

  • J. A. MiszczakEmail author
  • P. Gawron
  • Z. Puchała
Article

Abstract

We study a quantum version of a penny flip game played using control parameters of the Hamiltonian in the Heisenberg model. Moreover, we extend this game by introducing auxiliary spins which can be used to alter the behaviour of the system. We show that a player aware of the complex structure of the system used to implement the game can use this knowledge to gain higher mean payoff.

Keywords

Quantum information Quantum games Quantum control Heisenberg model Spin chain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

Work by P. Gawron was supported by the Polish Ministry of Science and Higher Education under the grant number IP2010 009770, J. A. Miszczak was supported by Polish National Science Centre under the project number N N516 475440, Z. Puchała was supported by the Polish National Science Centre under the research project N N514 513340.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Abal, G., Donangelo, R., Fort, H.: Conditional strategies in iterated quantum games. Phys. A Stat. Mech. its Appl. 387(21), 5326–5332 (2008). doi: 10.1016/j.physa.2008.04.036. http://www.sciencedirect.com/science/article/pii/S037843710800407X
  2. 2.
    Chandrashekar C., Banerjee S.: Parrondo’s game using a discrete-time quantum walk. Phys. Lett. A 375(14), 1553–1558 (2011). doi: 10.1016/j.physleta.2011.02.071 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen L., Ang H., Kiang D., Kwek L., Lo C.: Quantum prisoner dilemma under decoherence. Phys. Lett. A 316(5), 317–323 (2003)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Collins B., Śniady P.: Integration with respect to the haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773–795 (2006). doi: 10.1007/s00220-006-1554-3 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D’Alessandro D.: Introduction to Quantum Control and Dynamics. Chapman & Hall, London (2008)zbMATHGoogle Scholar
  6. 6.
    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(13), 137–902 (2002)CrossRefGoogle Scholar
  7. 7.
    Eisert J., Wilkens M., Lewenstein M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83(15), 3077–3080 (1999). doi: 10.1103/PhysRevLett.83.3077 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Flitney A.P., Abbott D.: Quantum games with decoherence. J. Phys. A Math. Gen. 38(2), 449–459 (2005). doi: 10.1088/0305-4470/38/2/011 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gawron P.: Noisy quantum Monty Hall game. Fluct. Noise Lett. 09(01), 9–18 (2010). doi: 10.1142/S0219477510000034 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gawron, P., Klamka, J., Winiarczyk, R.: Noise effects in quantum search algorithm from the computational complexity point of view. Int. J. Appl. Math. Comput. Sci. (2011), in pressGoogle Scholar
  11. 11.
    Gawron, P., Miszczak, J., Sładkowski, J.: Noise effects in quantum magic squares game. Int. J. Quantum Inf. 06(6 Supp 1), 667–673 (2008). doi: 10.1142/S0219749908003931 CrossRefzbMATHGoogle Scholar
  12. 12.
    Heule R., Bruder C., Burgarth D., Stojanović V.: Local quantum control of Heisenberg spin chains. Phys. Rev. A 82(5), 52–333 (2010). doi: 10.1103/PhysRevA.82.052333 CrossRefGoogle Scholar
  13. 13.
    Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001). http://www.scipy.org/
  14. 14.
    Meyer D.A.: Quantum strategies. Phys. Rev. Lett. 82(5), 1052–1055 (1999). doi: 10.1103/PhysRevLett.82.1052 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nocedal J., Wright S.: Numerical Optimization. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Pakuła I.: Quantum gambling using mesoscopic ring qubits. Phys. Stat. Sol. B 244(7), 2513–2515 (2007). doi: 10.1002/pssb.200674643 ADSCrossRefGoogle Scholar
  17. 17.
    Piotrowski E.W., Sładkowski J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003). doi: 10.1023/A:1025443111388 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ramzan M., Khan M.: Distinguishing quantum channels via magic squares game. Quantum Inf. Process. 9(6), 667–679 (2010). doi: 10.1007/s11128-009-0155-4 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Roy A., Scott A.: Unitary designs and codes. Des. Codes Cryptogr. 53(1), 13–31 (2009). doi: 10.1007/s10623-009-9290-2 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schmid C., Flitney A., Wieczorek W., Kiesel N., Weinfurter H., Hollenberg L.: Experimental implementation of a four-player quantum game. New J. Phys. 12, 31–63 (2010)Google Scholar
  21. 21.
    Zyczkowski K., Kuś M.: Random unitary matrices. J. Phys. A Math. Gen. 27(12), 4235 (1994). doi: 10.1088/0305-4470/27/12/028 ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute of Theoretical and Applied InformaticsPolish Academy of SciencesGliwicePoland

Personalised recommendations