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International Workshop on Quantum Technology and Optimization Problems

QTOP 2019: Quantum Technology and Optimization Problems pp 51–62Cite as

Nash Embedding and Equilibrium in Pure Quantum States

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11413))

Abstract

With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. We show here that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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References

  1. Albers, S., Eilts, S., Even-Dar, E., Mansour, Y., Roditty, L.: On Nash equilibria for a network creation game. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 89–98 (2006)

    Google Scholar 

  2. Fabrikant, A., Luthra, A., Menva, E., Papdimitriou, C., Shenker, S.: On a network creation game. In: Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing, pp. 347–351 (2003)

    Google Scholar 

  3. Liu, B., Dai, H., Zhang, M.: Playing distributed two-party quantum games on quantum networks. Quant. Inf. Process. 16, 290 (2017). https://doi.org/10.1007/s11128-017-1738-0

    Article  MathSciNet  MATH  Google Scholar 

  4. Scarpa, G.: Network games with quantum strategies. In: Sergienko, A., Pascazio, S., Villoresi, P. (eds.) QuantumComm 2009. LNICST, vol. 36, pp. 74–81. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11731-2_10

    Chapter  MATH  Google Scholar 

  5. Dasari, V., Sadlier, R.J., Prout, R., Williams, B.P., Humble, T.S.: Programmable multi-node quantum network design and simulation. In: SPIE Commercial+ Scientific Sensing and Imaging, pp. 98730B–98730B (2016)

    Google Scholar 

  6. Binmore, K.: Playing for Real. Oxford University Press, Oxford (2017)

    MATH  Google Scholar 

  7. Blaquiere, A.: Wave mechanics as a two-player game. In: Blaquiére, A., Fer, F., Marzollo, A. (eds.) Dynamical Systems and Microphysics, vol. 261, pp. 33–69. Springer, Vienna (1980). https://doi.org/10.1007/978-3-7091-4330-8_2

    Chapter  Google Scholar 

  8. Meyer, D.: Quantum strategies. Phys. Rev. Lett. 82, 1052–1055 (1999)

    Article  MathSciNet  Google Scholar 

  9. Eisert, J., Wilkens, M., Lewenstien, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077 (1999)

    Article  MathSciNet  Google Scholar 

  10. Nash, J.: Equilibrium points in N-player games. Proc. Natl. Acad. Sci. USA 36, 48–49 (1950)

    Article  Google Scholar 

  11. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8(3), 457–459 (1941)

    Article  MathSciNet  Google Scholar 

  12. Glicksberg, I.L.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proc. Am. Math. Soc. 3, 170–174 (1952)

    MathSciNet  MATH  Google Scholar 

  13. Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63(1), 20–63 (1956)

    Article  MathSciNet  Google Scholar 

  14. Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement, 1st edn. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  15. Browuer, L.: Ueber eineindeutige, stetige transformationen von Flächen in sich. Math. Ann. 69 (1910)

    Google Scholar 

  16. Caratheodory, C.: Math. Ann. 64, 95 (1907). https://doi.org/10.1007/BF01449883

  17. Kannai, Y.: An elementary proof of the no-retraction theorem. Am. Math. Mon. 88(4), 264–268 (1981)

    Article  MathSciNet  Google Scholar 

  18. Teklu, B., Olivares, S., Paris, M.: Bayesian estimation of one-parameter qubit gates. J. Phys. B: Atom. Mol. Opt. Phys. 42, 035502 (6pp) (2009)

    Google Scholar 

  19. Teklu, B., Genoni, M., Olivares, S., Paris, M.: Phase estimation in the presence of phase diffusion: the qubit case. Phys. Scr. T140, 014062 (3pp) (2010)

    Google Scholar 

  20. Khan, F.S., Solmeyer, N., Balu, R., Humble, T.S.: Quantum games: a review of the history, current state, and interpretation. Quant. Inf. Process. 17(11), 42 pp. Article ID 309. arXiv:1803.07919 [quant-ph]

  21. Khan, F.S., Phoenix, S.J.D.: Gaming the quantum. Quant. Inf. Comput. 13(3–4), 231–244 (2013)

    MathSciNet  Google Scholar 

  22. Khan, F.S., Phoenix, S.J.D.: Mini-maximizing two qubit quantum computations. Quant. Inf. Process. 12(12), 3807–3819 (2013)

    Article  MathSciNet  Google Scholar 

  23. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, p. 212, May 1996

    Google Scholar 

  24. Williams, B.P., Britt, K.A., Humble, T.S.: Tamper-indicating quantum seal. Phys. Rev. Appl. 5, 014001 (2016)

    Article  Google Scholar 

  25. Chitambar, E., Leung, D., Mančinska, L., Ozols, M., Winter, A.: Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys. 328(1), 303–326 (2014)

    Article  MathSciNet  Google Scholar 

  26. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution (preprint). https://arxiv.org/abs/quant-ph/0001106

  27. McGeoch, C.C.: Adiabatic Quantum Computation and Quantum Annealing. Morgan & Claypool Publishers Series. Morgan & Claypool Publishers, San Rafael (2014)

    Google Scholar 

  28. Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2, 5 (2014)

    Article  Google Scholar 

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Acknowledgments

Faisal Shah Khan is indebted to Davide La Torre and Joel Lucero-Bryan for helpful discussion on the topic of fixed-point theorems.

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Authors and Affiliations

  1. Center on Cyber-Physical Systems and Department of Mathematics, Khalifa University, Abu Dhabi, United Arab Emirates

    Faisal Shah Khan

  2. Quantum Computing Institute, Oak Ridge National Lab, Oak Ridge, TN, USA

    Travis S. Humble

Authors
  1. Faisal Shah Khan
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  2. Travis S. Humble
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Corresponding author

Correspondence to Faisal Shah Khan .

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Editors and Affiliations

  1. Ludwig-Maximilians-Universität München, Munich, Germany

    Sebastian Feld

  2. Ludwig-Maximilians-Universität München, Munich, Germany

    Claudia Linnhoff-Popien

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Khan, F.S., Humble, T.S. (2019). Nash Embedding and Equilibrium in Pure Quantum States. In: Feld, S., Linnhoff-Popien, C. (eds) Quantum Technology and Optimization Problems. QTOP 2019. Lecture Notes in Computer Science(), vol 11413. Springer, Cham. https://doi.org/10.1007/978-3-030-14082-3_5

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  • DOI: https://doi.org/10.1007/978-3-030-14082-3_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-14081-6

  • Online ISBN: 978-3-030-14082-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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