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.
Similar content being viewed by others
References
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
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
Chen L., Ang H., Kiang D., Kwek L., Lo C.: Quantum prisoner dilemma under decoherence. Phys. Lett. A 316(5), 317–323 (2003)
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
D’Alessandro D.: Introduction to Quantum Control and Dynamics. Chapman & Hall, London (2008)
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)
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
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
Gawron P.: Noisy quantum Monty Hall game. Fluct. Noise Lett. 09(01), 9–18 (2010). doi:10.1142/S0219477510000034
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 press
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
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
Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001). http://www.scipy.org/
Meyer D.A.: Quantum strategies. Phys. Rev. Lett. 82(5), 1052–1055 (1999). doi:10.1103/PhysRevLett.82.1052
Nocedal J., Wright S.: Numerical Optimization. Springer, New York (1999)
Pakuła I.: Quantum gambling using mesoscopic ring qubits. Phys. Stat. Sol. B 244(7), 2513–2515 (2007). doi:10.1002/pssb.200674643
Piotrowski E.W., Sładkowski J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003). doi:10.1023/A:1025443111388
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
Roy A., Scott A.: Unitary designs and codes. Des. Codes Cryptogr. 53(1), 13–31 (2009). doi:10.1007/s10623-009-9290-2
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)
Zyczkowski K., Kuś M.: Random unitary matrices. J. Phys. A Math. Gen. 27(12), 4235 (1994). doi:10.1088/0305-4470/27/12/028
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miszczak, J.A., Gawron, P. & Puchała, Z. Qubit flip game on a Heisenberg spin chain. Quantum Inf Process 11, 1571–1583 (2012). https://doi.org/10.1007/s11128-011-0322-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-011-0322-2