Advertisement

International Journal of Theoretical Physics

, Volume 58, Issue 6, pp 1968–1988 | Cite as

The Logos Categorical Approach to Quantum Mechanics: II. Quantum Superpositions and Intensive Values

  • C. de RondeEmail author
  • C. Massri
Article
  • 9 Downloads

Abstract

In this paper we attempt to consider quantum superpositions from the perspective of the logos categorical approach presented in de Ronde and Massri (30) . We will argue that our approach allows us not only to better visualize the structural features of quantum superpositions providing an anschaulich content to all terms, but also to restore —through the intensive valuation of graphs and the notion of immanent power— an objective representation of what QM is really talking about. In particular, we will discuss how superpositions relate to some of the main features of the theory of quanta, namely, contextuality, paraconsistency, probability and measurement.

Keywords

Categorical QM Logoi Quantum superpositions 

Notes

Acknowledgements

We would like to thank an anonymous referee for her/his careful reading, comments and corrections. This work was partially supported by the following grants: FWO project G.0405.08 and FWO-research community W0.030.06. CONICET RES. 4541-12 and the Project PIO-CONICET-UNAJ (15520150100008CO) “Quantum Superpositions in Quantum Information Processing”.

References

  1. 1.
    Adámek, J., Herrlich, H.: Cartesian closed categories, quasitopoi and topological universes. Comment. Math. Univ. Carol. 27, 235–257 (1986)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aerts, D.: A possible explanation for the probabilities of quantum mechanics. J. Math. Phys. 27, 202–210 (1986)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aerts, D.: A potentiality and conceptuality interpretation of quantum physics. Philosophica 83, 15–52 (2010)Google Scholar
  4. 4.
    Aerts, D., Sassoli de Bianchi, M.: Many-measurements or many-worlds? A dialogue. Found. Sci. 20, 399–427 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Allori, V.: Primitive ontology and the classical world. In: Kastner, R.E., Jeknic-Dugic, J., Jaroszkiewicz, G. (eds.) Quantum Structural Studies, pp. 175–199. World Scientific, Singapore (2016)Google Scholar
  6. 6.
    Altepeter, J.B., James, D.F., Kwiat, P.G.: 4 qubit quantum state tomography. In: Paris, M., Rehacek, J. (eds.) Quantum State Estimation. Springer, Berlin (2004)Google Scholar
  7. 7.
    Angot-Pellissier, R.: The relation between logic, set theory and topos 5 as it is used by Alain Badiouin. In: Koslow, A., Buchsbaum, A. (eds.) The Road to Universal Logic, pp. 181–200. Springer, Switzerland (2015)Google Scholar
  8. 8.
    Arenhart, J.R., Krause, D.: Oppositions in quantum mechanics. In: Béziau, J.-Y., Gan-Krzywoszynska, K. (eds.) New Dimensions of the Square of Opposition, pp. 337–356. Philosophia Verlag, Munich (2014)Google Scholar
  9. 9.
    Arenhart, J.R., Krause, D.: Potentiality and contradiction in quantum mechanics. In: Koslow, A., Buchsbaum, A. (eds.) The Road to Universal Logic, vol. II, pp. 201–211. Cham, Birkhäuser (2015)Google Scholar
  10. 10.
    Arenhart, J.R., Krause, D.: Contradiction, quantum mechanics, and the square of opposition. Logique et Analyse 59, 273–281 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bell, J.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 477 (1966)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bohr, N.: Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Butterfild, J.: Interview: Jeremy Butterfield: what is contextuality?. https://www.youtube.com/watch?v=ZJAnixX8T4U ZJAnixX8T4U (2017)
  14. 14.
    Cabello, A., Estebaranz, J.M., García-Alcaine, G.: Bell-Kochen-Specker theorem: a proof with 18 vectors. Phys. Lett. A 4, 183–187 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Carter, J.: Exploring the fruitfulness of diagrams in mathematics, Synthese,  https://doi.org/10.1007/s11229-017-1635-1. (philsci-archive:14130) (2017)
  16. 16.
    da Costa, N., de Ronde, C.: The paraconsistent logic of quantum superpositions. Found. Phys. 43, 845–858 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    da Costa, N., de Ronde, C.: The paraconsistent approach to quantum superpositions reloaded: formalizing contradictory powers in the potential realm. arXiv:quant-ph/1507.02706 (2015)
  18. 18.
    da Costa, N., de Ronde, C.: Revisiting the applicability of metaphysical identity in quantum mechanics. arXiv:quant-ph/1609.05361 (2016)
  19. 19.
    Dalla Chiara, M.L., Giuntini, R., Sergioli, G.: Probability in quantum computa-tiona and in quantum computational logic. Mathematical structures in computer science, 14. Cambridge University Press (2013)Google Scholar
  20. 20.
    de Ronde, C. In: Beziau, J.-Y., Chakraborty, M., Dutta, S. (eds.) : Modality, Potentiality and Contradiction in Quantum Mechanics, New Directions in Paraconsistent Logic. Springer, Berlin (2015)Google Scholar
  21. 21.
    de Ronde, C.: Probabilistic knowledge as objective knowledge in in quantum mechanics: potential immanent powers instead of actual properties. In: Aerts, D., de Ronde, C., Freytes, H., Giuntini, R. (eds.) Probing the Meaning of Quantum Mechanics: Superpositions, Semantics, Dynamics and Identity, pp. 141–178. World Scientific, Singapore (2016)Google Scholar
  22. 22.
    de Ronde, C.: Representational realism, closed theories and the quantum to classical limit. In: Kastner, R.E., Jeknic-Dugic, J., Jaroszkiewicz, G. (eds.) Quantum Structural Studies, pp. 105–135. World Scientific, Singapore (2016)Google Scholar
  23. 23.
    de Ronde, C.: Causality and the modeling of the measurement process in quantum theory. Disputatio 9, 657–690 (2017)CrossRefGoogle Scholar
  24. 24.
    de Ronde, C.: Quantum superpositions and the representation of physical reality beyond measurement outcomes and mathematical structures. Found. Sci. 23, 621–648 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    de Ronde, C.: Immanent powers versus causal powers (propensities, latencies and dispositions) in quantum mechanics. In: Aerts, D., Dalla Chiara, M.L., de Ronde, C., Krause, D. (eds.) Probing the Meaning of Quantum Mechanics: : Information, Contextuality, Relationalism and Entanglement, pp. 121–157. World Scientific, Singapore (2019)Google Scholar
  26. 26.
    de Ronde, C.: A defense of the paraconsistent approach to quantum superpositions (reply to Arenhart and Krause), Metatheoria, forthcoming (2019)Google Scholar
  27. 27.
    de Ronde, C.: Unscrambling the omelette of quantum contextuality (Part I): Preexistent Properties or Measurement Outcomes? Found. Sci.,  https://doi.org/10.1007/s10699-019-09578-8 (2019)
  28. 28.
    de Ronde, C.: Potential truth in quantum mechanics, (2019)Google Scholar
  29. 29.
    de Ronde, C., Massri, C.: Kochen-Specker Theorem, Physical Invariance and Quantum Individuality. Cadernos da Filosofia da Ciencia 2, 107–130 (2017)Google Scholar
  30. 30.
    de Ronde, C., Massri, C.: The logos categorical approach to quantum mechanics: I. Kochen-Specker contextuality and global intensive valuations., Int. J. Theor. Phys.,  https://doi.org/10.1007/s10773-018-3914-0. arXiv:quant-ph/1801.00446 (2018)
  31. 31.
    de Ronde, C., Freytes, H., Domenech, G.: Interpreting the modal Kochen-Specker theorem: possibility and many worlds in quantum mechanics. Stud. Hist. Philos. Mod. Phys. 45, 11–18 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    D’Espagnat, B.: Veiled Reality: an Analysis of Present-Day Quantum Mechanical Concepts. Westview Press, Colorado (2003)zbMATHGoogle Scholar
  33. 33.
    Deutsch, D.: The Fabric of Reality. Penguin, London (1997)Google Scholar
  34. 34.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  35. 35.
    Domenech, G., Freytes, H., de Ronde, C.: Scopes and limits of modality in quantum mechanics. Annalen der Physik 15, 853–860 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Eva, B. In: Aerts, D., de Ronde, C., Freytes, H., Giuntini, R. (eds.) : A topos theoretic framework for paraconsistent quantum theory, probing the meaning of quantum mechanics: superpositions, dynamics, Semantics and identity, pp. 340–350. World Scientific, Singapore (2016)Google Scholar
  37. 37.
    Fuchs, C.A., Peres, A.: Quantum theory needs no ‘interpretation’. Phys. Today 53, 70 (2000)CrossRefGoogle Scholar
  38. 38.
    Fuchs, C.A., Mermin, N.D., Schack, R.: An introduction to QBism with an application to the locality of quantum mechanics. Am. J. Phys. 82, 749 (2014). arXiv:quant-ph/1311.5253 ADSCrossRefGoogle Scholar
  39. 39.
    Grangier, P.: Contextual objectivity: a realistic interpretation of quantum mechanics. Eur. J. Phys. 23, 331 (2002)CrossRefGoogle Scholar
  40. 40.
    Griffiths, R.B.: Hilbert space quantum mechanics is non contextual. Stud. Hist. Philos. Mod. Phys. 44, 174–181 (2013)CrossRefzbMATHGoogle Scholar
  41. 41.
    Hawking, S., Penrose, R.: The Nature of Space and Time. Princeton University Press, Princeton (2015)zbMATHGoogle Scholar
  42. 42.
    Isham, C.J., Butterfield, J.: A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalised valuations. Int. J. Theor. Phys. 37, 2669–2733 (1998)CrossRefzbMATHGoogle Scholar
  43. 43.
    Karakostas, V., Zafiris, E.: Contextual semantics in quantum mechanics from a categorical point of view. Synthese 194, 847–886 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kastner, R.: The Transactional Interpretation of Quantum Mechanics: the Reality of Possibility. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  45. 45.
    Krause, D., Arenhart, J. In: Aerts, D., de Ronde, C., Freytes, H., Giuntini, R. (eds.) : A logical account of superpositions, in probing the meaning and structure of quantum mechanics: superpositions, semantics, dynamics and identity, pp. 44–59. World Scientific, Singapore (2015)Google Scholar
  46. 46.
    Ledda, A., Sergioli, G.: Towards quantum computational logics. Int. J. Theo. Phys. 49, 3158–3165 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)zbMATHGoogle Scholar
  48. 48.
    Penrose, R.: Interview series: many worlds of quantum theory. In: Closer to Truth. https://www.closertotruth.com/series/many-worlds-quantum-theory closertotruth.com/series/many-worlds-quantum-theory (1998)
  49. 49.
    Pitowsky, I.: George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Br. J. Philos. Sci. 45, 95–125 (1994)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Schrödinger, E.: The present situation in Quantum Mechanics, Naturwiss, 23, 807–812. Translated to english in Quantum Theory and Measurement, Wheeler, J.A. and Zurek, W.H. (eds), 1983, Princeton University Press, Princeton (1935)Google Scholar
  51. 51.
    Steane, A.M.: A quantum computer only needs one universe. Studies in History and Philosophy of Modern Physics B 34, 469–478 (2003). arXiv:quant-ph/0003084v1 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Suárez, M.: Quantum propensities. Studies in History and Philosophy of Modern Physics 38, 418–438 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Svozil, K.: Classical versus quantum probabilities and correlations, arXiv:quant-ph/1707.08915 (2017)
  54. 54.
    Wallace, Interview with David. In: Schlosshauer, M. (ed.) : Elegance and Enigma: the Quantum Interviews. Springer, Berlin (2011)Google Scholar
  55. 55.
    Wallace, D.: The Emergent Multiverse: Quantum Theory According to the Everett. Interpretation. Oxford University Press, Oxford (2012)CrossRefzbMATHGoogle Scholar
  56. 56.
    Wallace, D.: Interview series: many worlds of quantum theory. In: Closer to Truth. https://www.closertotruth.com/series/many-worlds-quantum-theory closertotruth.com/series/many-worlds-quantum-theory (2012)
  57. 57.
    Weyl, H.: Philosophy of Mathematics and Natural Science (Revised and Augmented English Edition Based on a Translation by Olaf Helmer). Princeton University Press, Princeton (1949)zbMATHGoogle Scholar
  58. 58.
    Wheeler, J., Zurek, W.: Theory and Measurement. Princeton University Press, Princeton (1983)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Philosophy Institute Dr. A. KornUniversity of Buenos Aires - CONICETBuenos AiresArgentina
  2. 2.Center Leo Apostel for Interdisciplinary StudiesFoundations of the Exact Sciences - Vrije Universiteit BrusselBrusselBelgium
  3. 3.Institute of EngineeringNational University Arturo JauretcheBuenos AiresArgentina
  4. 4.Institute of Mathematical Investigations Luis A. Santaló, UBA - CONICETBuenos AiresArgentina
  5. 5.University CAECEBuenos AiresArgentina

Personalised recommendations