Skip to main content
SpringerLink
Log in

Quantum Entanglement on a Hypersphere

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

A quantum entanglement’s composite system does not display separable states and a single constituent cannot be fully described without considering the other states. We introduce quantum entanglement on a hypersphere - which is a 4D space undetectable by observers living in a 3D world -, derived from signals originating on the surface of an ordinary 3D sphere. From the far-flung branch of algebraic topology, the Borsuk-Ulam theorem states that, when a pair of opposite (antipodal) points on a hypersphere are projected onto the surface of 3D sphere, the projections have matching description. In touch with this theorem, we show that a separable state can be achieved for each of the entangled particles, just by embedding them in a higher dimensional space. We view quantum entanglement as the simultaneous activation of signals in a 3D space mapped into a hypersphere. By showing that the particles are entangled at the 3D level and un-entangled at the 4D hypersphere level, we achieved a composite system in which each local constituent is equipped with a pure state. We anticipate this new view of quantum entanglement leading to what are known as qubit information systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Jaeger, G.: Quantum Information an Overview. Springer, New York (2007)

    MATH  Google Scholar 

  2. Gruska, J.: Quantum entanglement as a new information processing resource. N. Gener. Comput. 21, 279–295 (2003)

    Article  MATH  Google Scholar 

  3. Hauke, P., Tagliacozzo, L.: Spread of correlations in long-range interacting quantum systems. Phys. Rev. Lett. 111(20), 207202 (2013)

    Article  ADS  Google Scholar 

  4. Horodecki, P., Horodecki, M.: Entanglement and thermodynamical analogies. Acta Physica Slovaca 3, 141–56 (1998)

    MATH  Google Scholar 

  5. Borsuk, K.: Drei saetze ueber die n-dimensionale euklidische sphaere. Fundam. Math. XX, 177–190 (1933)

    MATH  Google Scholar 

  6. Cohen, M.M.: A Course in Simple Homotopy Theory. Springer, New York (1973)

    Book  MATH  Google Scholar 

  7. Schroedinger, E.: Discussion of probabilty relations between separated systems. Proc. Camb. Philos. Soc. 31, 555–563 (1935)

    Article  ADS  Google Scholar 

  8. Peters, J.F.: Computional Proximity. Excursions in the Topology of Digital Images, Intelligent Systems (Reference Library 102, Springer, 2015, doi:10.1007/978-3-319-30262-1)

  9. Weisstein, E.W.: Antipodal points (http://mathworld.wolfram.com/AntipodalPoints.html. 2015)

  10. Henderson, D.W.: Experiencing Geometry. In: Euclidean, Spherical, and Hyperbolic Spaces. 2nd edn. (2001)

  11. Naimpally, S.A., Peters, J.F.: Preservation of continuity. Scientiae Mathematicae Japonicae 76, 305–311 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Willard, S.: General Topology. Dover Pub., Inc., Mineola (1970)

    MATH  Google Scholar 

  13. Collins, G.P.: The shapes of space. Sci. Am. 291, 94–103 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  14. Borsuk, M., Gmurczyk, A.: On homotopy types of 2-dimensional polyhedral. Fundam. Math. 109, 123–142 (1980)

    MathSciNet  MATH  Google Scholar 

  15. Dodson, C.T.J., Parker, P.E.: A User’s guide to algebraic topology. Kluwer, Dordrecht (1997)

    Book  MATH  Google Scholar 

  16. Leinaas, J.M., et al.: Geometrical aspects of entanglement, arXiv:0605079v1, pp. 1–31 (2006)

  17. Borsuk, M.: Concerning the classification of topological spaces from the standpoint of the theory of retracts. Fundam. Math. XLVI, 177–190 (1958)

    MathSciNet  Google Scholar 

  18. Borsuk, M.: Fundamental retracts and extensions of fundamental sequences 64, 55–85 (1969)

    MathSciNet  Google Scholar 

  19. Krantz, S.G.: A Guide to Topology. The Mathematical Association of America, Washington (2009)

    MATH  Google Scholar 

  20. Manetti, M.: Topology. Springer, Heidelberg (2015)

    Book  MATH  Google Scholar 

  21. Peters, J.F.: Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces. Intelligent systems (reference library, 63 Springer, 2014)

  22. Weisstein, E.W.: Pathwise-connected (Mathworld, http://mathworld.wolfram.com/Pathwise-Connected.html. 2015)

  23. Zhang, Y., et al.: Simulating quantum state engineering in spontaneous parametric down-conversion using classical light. Opt. Express. 22, 17039–17049 (2014)

    Article  ADS  Google Scholar 

  24. Quesada, N., Sipe, J.E.: Time-ordering effects in the generation of entangled photons using nonlinear optical processes. Phys. Rev. Lett. 114, 093903 (2015)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Manitoba, 75A Chancellor’s Circle, R3T 5V6, Winnipeg, MB, Canada

    James F. Peters

  2. Center for Nonlinear Science, University of North Texas, 1155 Union Circle, #311427, Denton, TX, 76203-5017, USA

    Arturo Tozzi

Authors
  1. James F. Peters
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arturo Tozzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arturo Tozzi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peters, J.F., Tozzi, A. Quantum Entanglement on a Hypersphere. Int J Theor Phys 55, 3689–3696 (2016). https://doi.org/10.1007/s10773-016-2998-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-016-2998-7

Keywords

Search

Navigation