Advertisement

Entropy, Information, Chaos and the Quantum Potential

  • Ignazio LicataEmail author
  • Davide Fiscaletti
Chapter
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)

Abstract

According to the modern research, several important results have been obtained as regards quantum entropy and quantum information. In this chapter we analyse the several approaches to quantum entropy (von Neumann, Kak, Sbitnev) and finally a geometric approach suggested recently by the authors of this book.

Keywords

Pure State Fisher Information Quantum Potential Chaotic Orbit Shannon Information 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Lieb, E.: Proof of an entropy conjecture of Wehrl. Commun. Math. Phys. 62, 35–41 (1978)MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Kak, S.: Quantum information and entropy. Int. J. Theor. Phys. 46(4), 860–876 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Frieden, B.R.: Science from Fisher information: a unification. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  4. 4.
    Hall, J.W.M.: Quantum properties of classical Fisher information. Phys. Rev. A 62(1), 012107 (2000)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Shun-Long, L.: Fisher information of wavefunctions: classical and quantum. Chin. Phys. Lett 23(12), 3127–3130 (2006)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Luati, A.: Maximum Fisher information in mixed state quantum systems. Ann. Stat. 32(4), 1770–1779 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fiscaletti, D., Licata, I.: Weyl geometry, Fisher information and quantum entropy in quantum mechanics. Inter. J. Theor. Phys. 51(11), 3587–3595 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Resconi, G., Licata, I., Fiscaletti, D.: Unification of quantum and gravity by non classical information entropy space. Entropy 15, 3602–3619 (2013)CrossRefGoogle Scholar
  9. 9.
    Castro, C., Mahecha, J.: On nonlinear quantum mechanics, Brownian motion, Weyl geometry and Fisher information. Prog. Phys. 1, 38–45 (2006)Google Scholar
  10. 10.
    Licata, I.: Effective physical processes and active information in quantum computing. Quant. Biosyst. 1, 51–65 (2007) (Reprinted in New Trends in quantum information, Licata, I., Sakaji, A., Singh, J.P., Felloni, S. (eds.) Aracne Publ, Rome (2010))Google Scholar
  11. 11.
    Licata, I.: Beyond turing: hypercomputation and quantum morphogenesis. Asia Pac. Math. Newslett. 2(3), 20–24 (2012)MathSciNetGoogle Scholar
  12. 12.
    Dürr, D., Goldstein, S., Zanghi, N.: Quantum chaos, classical randomness, and Bohmian mechanics. J. Stat. Phys. 68, 259–270 (1992)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Schwengelbeck, U., Faisal, F.H.M.: Definition of Lyapunov exponents and KS entropy in quantum dynamics. Phys. Lett. A 199, 281–286 (1995)MathSciNetADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Parmenter, R.H., Valentine, R.W.: Deterministic chaos and the causal interpretation of quantum mechanics. Phys. Lett. A 201, 1–8 (1995)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Faisal, F.H.M., Schwengelbeck, U.: Unified theory of Lyapunov exponents and a positive example of deterministic quantum chaos. Phys. Lett. A 207, 31–36 (1995)Google Scholar
  16. 16.
    Iacomelli, G., Pettini, M.: Regular and chaotic quantum motion. Phys. Lett. A 212, 29–38 (1996)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Sengupta, S., Chattaraj, P.K.: The quantum theory of motion and signature of chaos in the quantum behavior of a classically chaotic system. Phys. Lett. A 215, 119–127 (1996)ADSCrossRefGoogle Scholar
  18. 18.
    Parmenter, R.H., Valentine, R.W.: Properties of the geometric phase of a de Broglie-Bohm causal quantum mechanical trajectory. Phys. Lett. A 219, 7–14 (1996)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Dewdney, C., Malik, Z.: Measurement, decoherence and chaos in quantum pinball. Phys. Lett. A 220, 183–188 (1996)MathSciNetADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Oriols, X., Martìn, F., Suñé′, J.: Implications of the noncrossing property of Bohm trajectories in one-dimensional tunneling configurations. Phys. Rev. A 54, 2594–2604 (1996)Google Scholar
  21. 21.
    Parmenter, R.H., Valentine, R.W.: Chaotic causal trajectories associated with a single stationary state of a system of non-interacting particles. Phys. Lett. A 227, 5–14 (1997)MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    De Alcantara Bonfim, O.F., Florencio, J., Sa Barreto, F.C.: An approach based on Bohm’s view of quantum mechanics quantum chaos in a double square well. Phys. Rev. E 58, 6851–6854 (1998)Google Scholar
  23. 23.
    Goldstein, S.: Absence of chaos in Bohmian dynamics. Phys. Rev. E 60, 7578–7579 (1999)ADSCrossRefGoogle Scholar
  24. 24.
    Wu, H., Sprung, D.W.L.: Quantum chaos in terms of Bohm trajectories. Phys. Lett. A 261, 150–157 (1999)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Wisniacki, D., Pujals, E.: Motion of vortices implies chaos in Bohmian mechanics. Europhys. Lett. 71, 159 (2005)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Efthymiopoulos, C., Contopoulos, G.: Chaos in Bohmian quantum mechanics. J. Phys. A: Math. Gen. 39, 1819–1852 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Efthymiopoulos, C., Kalapotharakos, C., Contopoulos, G.: Nodal points and the transition from ordered to chaotic Bohmian trajectories. J. Phys. A 40, 12945–12972 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Efthymiopoulos, C., Kalapotharakos, C., Contopoulos, G.: Origin of chaos near critical points of quantum flow. Phys. Rev. E. Stat Nonlin. Soft. Matter Phys 79, 1–18 (2009)MathSciNetGoogle Scholar
  29. 29.
    Contopoulos, G., Delis, N., Efthymiopoulos, C.: Order in de Broglie-Bohm mechanics. J. Phys. A.: Math. Theor. 45, 5301 (2012)Google Scholar
  30. 30.
    Parmenter, R.H., DiRienzo, A.L.: Reappraisal of the causal interpretation of quantum mechanics and of the quantum potential concept arXiv:quant-ph/0305183 (2003) Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Institute for Scientific MethodologyPalermoItaly

Personalised recommendations