International Journal of Theoretical Physics

, Volume 52, Issue 3, pp 723–734 | Cite as

Epistemic Entanglement due to Non-generating Partitions of Classical Dynamical Systems

  • Peter beim Graben
  • Thomas Filk
  • Harald Atmanspacher


Quantum entanglement relies on the fact that pure quantum states are dispersive and often inseparable. Since pure classical states are dispersion-free they are always separable and cannot be entangled. However, entanglement is possible for epistemic, dispersive classical states. We show how such epistemic entanglement arises for epistemic states of classical dynamical systems based on phase space partitions that are not generating. We compute epistemically entangled states for two coupled harmonic oscillators.


Entanglement Classical dynamical systems Partitions Coupled oscillators 



This research was partly supported by the Franklin Fetzer Fund and by a DFG Heisenberg grant awarded to PbG (GR 3711/1-1).


  1. 1.
    Allahverdyan, A.E., Khrennikov, A., Nieuwenhuizen, T.M.: Brownian entanglement. Phys. Rev. A 72, 032102 (2005) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Atmanspacher, H.: Quantum approaches to consciousness. In: Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy (2011). Google Scholar
  3. 3.
    Atmanspacher, H., Primas, H.: Epistemic and ontic quantum realities. In: Castell, L., Ischebeck, O. (eds.) Time, Quantum and Information, pp. 301–321. Springer, Berlin (2003) Google Scholar
  4. 4.
    Atmanspacher, H., Filk, T., beim Graben, P.: Can classical epistemic states be entangled? In: Song, D., et al. (eds.) Quantum Interaction—QI 2011, pp. 105–115. Springer, Berlin (2011) CrossRefGoogle Scholar
  5. 5.
    Bollt, E.M., Stanford, T., Lai, Y.C., Życzkowski, K.: What symbolic dynamics do we get with a misplaced partition? On the validity of threshold crossings analysis of chaotic time-series. Physica D 154, 259–286 (2001) MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Bruza, P.D., Kitto, K., Nelson, D., McEvoy, C.L.: Is there something quantum-like about the human mental lexicon? J. Math. Psychol. 53, 362–377 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Springer, Berlin (1982) MATHCrossRefGoogle Scholar
  8. 8.
    Dvurečenskij, A., Pulmannová, S., Svozil, K.: Partition logics, orthoalgebras and automata. Helv. Phys. Acta 68, 407–428 (1995) MathSciNetMATHGoogle Scholar
  9. 9.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777–780 (1935) ADSMATHCrossRefGoogle Scholar
  10. 10.
    Foulis, D.J.: A half-century of quantum logic. What have we learned? In: Aerts, D. (ed.) Quantum Structures and the Nature of Reality, Einstein Meets Magritte: An Interdisciplinary Reflection on Science, Nature, Art, Human Action and Society, vol. 7, pp. 1–36. Kluwer Academic, Dordrecht (1999) Google Scholar
  11. 11.
    van Gelder, T.: The dynamical hypothesis in cognitive science. Brain Behav. Sci. 21, 1–14 (1998) Google Scholar
  12. 12.
    beim Graben, P., Atmanspacher, H.: Complementarity in classical dynamical systems. Found. Phys. 36, 291–306 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    beim Graben, P., Atmanspacher, H.: Extending the philosophical significance of the idea of complementarity. In: Atmanspacher, H., Primas, H. (eds.) Recasting Reality. Wolfgang Pauli’s Philosophical Ideas and Contemporary Science, pp. 99–113. Springer, Berlin (2009) CrossRefGoogle Scholar
  14. 14.
    Khrennikov, A.Y.: Representation of the Kolmogorov model having all distinguishing features of quantum probabilistic model. Phys. Lett. A 316(5), 279–296 (2003) MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Khrennikov, A.Y.: Ubiquitous Quantum Structure: From Psychology to Finance. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  16. 16.
    Kolmogorov, A.N.: New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces. Dokl. Russ. Acad. Sci. 119, 861–864 (1958) MathSciNetMATHGoogle Scholar
  17. 17.
    Lind, D.A., Marcus, B.: An Introduction to Symbolic Dynamics. Cambridge University Press, Cambridge (1995) MATHCrossRefGoogle Scholar
  18. 18.
    Pauli, W.: Die philosophische Bedeutung der Idee der Komplementarität. Experientia 6, 72–81 (1950). English translation: The philosophical significance of the notion of complementarity. In: Enz, C.P., von Meyenn, K. (eds.) Wolfgang Pauli. Writings on Physics and Philosophy, pp. 35–42 Springer, Berlin (1994) CrossRefGoogle Scholar
  19. 19.
    Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Camb. Philos. Soc. 31, 555–563 (1935) ADSCrossRefGoogle Scholar
  20. 20.
    Sinai, Y.G.: On the concept of entropy of a dynamical system. Dokl. Russ. Acad. Sci. 124, 768–771 (1959) MathSciNetMATHGoogle Scholar
  21. 21.
    Svozil, K.: Logical equivalence between generalized urn models and finite automata. Int. J. Theor. Phys. 44, 745–754 (2005) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Walters, P.: Introduction to Ergodic Theory. Springer, Berlin (1981) Google Scholar
  23. 23.
    Wright, R.: Generalized urn models. Found. Phys. 20, 881–903 (1990) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Peter beim Graben
    • 1
  • Thomas Filk
    • 2
  • Harald Atmanspacher
    • 3
  1. 1.Dept. of German Language and Linguistics, and Bernstein Center for Computational NeuroscienceHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Physikalisches InstitutUniversity of FreiburgFreiburg i. Br.Germany
  3. 3.Institut für Grenzgebiete der Psychologie und PsychohygieneFreiburg i. Br.Germany

Personalised recommendations