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Quantum-Theoretic Modeling in Computer Science

A Complex Hilbert Space Model for Entangled Concepts in Corpuses of Documents
  • Diederik Aerts
  • Lester Beltran
  • Suzette Geriente
  • Sandro SozzoEmail author
Article

Abstract

We work out a quantum-theoretic model in complex Hilbert space of a recently performed test on co-occurrencies of two concepts and their combination in retrieval processes on specific corpuses of documents. The test violated the Clauser-Horne-Shimony-Holt version of Bell’s inequalities (‘CHSH inequality’), thus indicating the presence of entanglement between the combined concepts. We make use of a recently elaborated ‘entanglement scheme’ and represent the collected data in the tensor product of Hilbert spaces of the individual concepts, showing that the identified violation is due to the presence of a strong form of entanglement, involving both states and measurements and reflecting the meaning connection between the component concepts. These results provide a significant confirmation of the presence of quantum structures in corpuses of documents, like it is the case for the entanglement identified in human cognition.

Keywords

CHSH inequality Quantum entanglement Quantum structures Corpuses of documents Natural language processing 

Notes

Acknowledgements

This work was supported by QUARTZ (Quantum Information Access and Retrieval Theory), the Marie Skłodowska-Curie Innovative Training Network 721321 of the European Union’s Horizon 2020 research and innovation programme.

References

  1. 1.
    Aerts, D.: Quantum structure in cognition. J. Math. Psychol. 53, 314–348 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Khrennikov, A.: Ubiquitous Quantum Structure. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Busemeyer, J., Bruza, P.: Quantum Models of Cognition and Decision. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  4. 4.
    Aerts, D., Broekaert, J., Gabora, L., Sozzo, S.: Quantum structure and human thought. Behav. Brain. Sci. 36, 274–276 (2013)CrossRefGoogle Scholar
  5. 5.
    Aerts, D., Gabora, L., Sozzo, S.: Concepts and their dynamics: A quantum theoretic modeling of human thought. Top. Cogn. Sci. 5, 737–772 (2013)Google Scholar
  6. 6.
    Kvam, P., Pleskac, T., Yu, S., Busemeyer, J.: Interference effects of choice on confidence. In: Proceedings of the National Academy of Science of the USA112, pp. 10645–10650 (2015)Google Scholar
  7. 7.
    Dalla Chiara, M.L., Giuntini, R., Negri, E.: A quantum approach to vagueness and to the semantics of music. Int. J. Theor. Phys. 54, 4546–4556 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dalla Chiara, M.L., Giuntini, R., Leporini, R., Negri, E., Sergioli, G.: Quantum information, cognition, and music. Front. Psychol. 6, 1583 (2015)CrossRefGoogle Scholar
  9. 9.
    Aerts, D., Haven, E., Sozzo, S.: A proposal to extend expected utility in a quantum probabilistic framework. Economic. Theory. 65, 1079–1109 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Schmitt, I.D., Nurnberger, A.: Towards quantum logic based multimedia retrieval. NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.  https://doi.org/10.1109/NAFIPS.2008.4531329 (2008)
  11. 11.
    Bruza, P., Kitto, K., Nelson, D., McEvoy, C.: Is there something quantum-like about the human mental lexicon? J. Math. Psychol. 53, 362–377 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coecke, B., Sadrzadeh, M., Clark, S.: Mathematical foundations for a compositional distributional model of meaning. Linguist. Anal. 36, 345–384 (2010)Google Scholar
  13. 13.
    Piwowarski, B., Frommholz, I., Lalmas, M., van Rijsbergen, K.: What can quantum theory bring to information retrieval. In: Proceedings of the 19th ACM International Conference on Information and Knowledge Management, pp. 59–68.  https://doi.org/10.1145/1871437.1871450 (2010)
  14. 14.
    Frommholz, I., Larsen, B., Piwowarski, B., Lalmas, M., Ingwersen, P.: Supporting polyrepresentation in a quantum-inspired geometrical retrieval framework. In: IIiX ’10 Proceedings of the Third Symposium on Information Interaction in Context, pp. 115–124.  https://doi.org/10.1145/1840784.1840802 (2010)
  15. 15.
    Zellhöfer, D., Frommholz, I., Schmitt, I., Lalmas, M., van Rijsbergen, K.: Towards quantum-based DB+IR processing based on the principle of polyrepresentation. In: Clough, P., et al. (eds.) Advances in Information Retrieval, ECIR 2011, LNCS 6611, pp. 729–732. Springer, Berlin (2011)Google Scholar
  16. 16.
    Di Buccio, E., Melucci, M., Song, D.: Towards predicting relevance using a quantum-like framework. In: Clough, P. et al. (eds.) Advances in Information Retrieval. ECIR 2011. Lecture Notes in Computer Science, vol. 6611, pp. 755–758. Springer, Berlin (2011)Google Scholar
  17. 17.
    Melucci, M.: Introduction to Information Retrieval and Quantum Mechanics. Springer, Berlin Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Aerts, D., Arguelles, J., Beltran, L., Beltran, L., Sassoli de Bianchi, M., Sozzo, S., Veloz, T.: Testing quantum models of conjunction fallacy on the world wide web. Int. J. Theor. Phys. 56, 3744–3756 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Aerts, D., Aerts Arguëlles, J., Beltran, L., Beltran, L., Distrito, I., Sassoli de Bianchia, M., Sozzo, S., Veloz, T.: Towards a quantum world wide web. Theor. Comput. Sci. 752, 116–131 (2019)Google Scholar
  20. 20.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 31, 555–563 (1935)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Bohm, D.: Quantum Theory. New-York, Prentice-Hall (1951)Google Scholar
  23. 23.
    Bell, J.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Bell, J.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  25. 25.
    Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Aspect, A., Grangier, G., Roger, G.: Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: A new violation of Bell’s Inequalities. Phys. Rev. Lett. 49, 91–94 (1982)ADSCrossRefGoogle Scholar
  27. 27.
    Yin, J., et al.: Satellite-based entanglement distribution over 1200 kilometers. Science 356, 1140–1144 (2017).  https://doi.org/10.1126/science.aan3211 CrossRefGoogle Scholar
  28. 28.
    Bruza, P., Kitto, K., Nelson, D., McEvoy, C.: Extracting spooky-activation-at-a-distance from considerations of entanglement. In: Bruza, P., Sofge, D., Lawless, W., van Rijsbergen K., Klusch, M. (eds.) Quantum Interaction. QI 2009. Lecture Notes in Computer Science, vol. 5494, pp. 71–83. Springer, Berlin (2009)Google Scholar
  29. 29.
    Aerts, D., Sozzo, S.: Quantum structure in cognition: Why and how concepts are entangled. Quant. Interact. Lect. Notes. Comput. Sci. 7052, 116–127 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Aerts, D., Sozzo, S.: Quantum entanglement in concept combinations. Int. J. Theor. Phys. 53, 3587–3603 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Bruza, P., Kitto, K., Ramm, B., Sitbon, L.: A probabilistic framework for analysing the compositionality of conceptual combinations. J. Math. Psychol. 67, 26–38 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Gronchi, G., Strambini, E.: Quantum cognition and Bell’s inequality: A model for probabilistic judgment bias. J. Math. Psychol. 78, 65–75 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Aerts, D., Aerts Arguëlles, J., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S., Veloz, T.: Spin and wind directions I: Identifying entanglement in nature and cognition. Found. Sci. 23, 323–335 (2018)CrossRefGoogle Scholar
  34. 34.
    Aerts, D., Aerts Arguëlles, J., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S., Veloz, T.: Spin and wind directions II: A Bell state quantum model. Found. Sci. 23, 337–365 (2018)CrossRefzbMATHGoogle Scholar
  35. 35.
    Beltran, L., Geriente, S.: Quantum entanglement in corpuses of documents. Found. Sci. 24, 227–246 (2019) Google Scholar
  36. 36.
    Cirel’son, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93–100 (1980)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Aerts, D., Aerts Arguëlles, J., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S., Veloz, T.: Quantum entanglement in physical and cognitive systems: A conceptual analysis and a general representation. arXiv:1903.09103 [q-bio] (2019)
  38. 38.
    Aerts, D., Sassoli de Bianchi, M., Sozzo, S.: On the foundations of the Brussels operational-realistic approach to cognition. Front. Phys. 4, 17 (2016).  https://doi.org/10.3389/fphy.2016.00017 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Diederik Aerts
    • 1
  • Lester Beltran
    • 1
  • Suzette Geriente
    • 1
  • Sandro Sozzo
    • 2
    Email author
  1. 1.Center Leo Apostel (Clea)Free University of Brussels (VUB)BrusselBelgium
  2. 2.School of Business and Centre IQSCSUniversity of LeicesterLeicesterUK

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