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How quantum mechanics might be when measuring commuting observables

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Abstract

We consider how quantum mechanics might be when measuring commuting observables if we rely on the property of the Kronecker delta. Quantum mechanics is reduced to a classical theory when we consider only commuting observables. Using this fact, we discuss an inconsistency within quantum mechanics when accepting we use the property of the Kronecker delta without extra assumptions about the reality of observables. One of the objectives of this paper is for us to remain wondering the extension of quantum mechanical axiom to concrete commuting observables themselves.

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References

  1. J.J. Sakurai, Modern quantum mechanics, Revised. (Addison-Wesley Publishing Company, Boston, 1995)

    Google Scholar 

  2. A. Peres, Quantum theory: concepts and methods (Kluwer Academic, Dordrecht, 1993)

    MATH  Google Scholar 

  3. M. Redhead, Incompleteness, nonlocality, and realism, 2nd edn. (Clarendon Press, Oxford, 1989)

    MATH  Google Scholar 

  4. J. von Neumann, Mathematical foundations of quantum mechanics (Princeton University Press, Princeton, 1955)

    MATH  Google Scholar 

  5. M. A. Nielsen, I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000)

  6. A. S. Holevo, Quantum systems, channels, information, a mathematical introduction (De Gruyter, 2012). https://doi.org/10.1515/9783110273403

  7. K. Nagata, D. N. Diep, A. Farouk, T. Nakamura, Simplified quantum computing with applications (IOP Publishing, Bristol, UK, 2022). https://doi.org/10.1088/978-0-7503-4700-6

  8. A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)

    Article  ADS  MATH  Google Scholar 

  9. S. Kochen, E.P. Specker, The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1967)

    MathSciNet  MATH  Google Scholar 

  10. K. Nagata, S.K. Patro, T. Nakamura, The Kochen–Specker theorem based on the kronecker delta. Int. J. Theor. Phys. 58, 1311 (2019). https://doi.org/10.1007/s10773-019-04023-9

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Clapham, J. Nicholson, The concise oxford dictionary of mathematics, 5th edn. (Oxford University Press, Oxford, 2014)

    MATH  Google Scholar 

  12. G. W. Mackey, “The mathematical foundations of quantum mechanics” (Dover Books on Physics) Illustrated Edition (1963)

  13. S. Gudder, On the quantum logic approach to quantum mechanics. Commun. math. Phys. 12, 1–15 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. W. Guz, A modification of the axiom system of quantum mechanics. Rep. Math. Phys. 7(2), 313–320 (1975). https://doi.org/10.1016/0034-4877(75)90036-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. W. Guz, Conditional probability and the axiomatic structure of quantum mechanics. Fortschritte der Physik 29(8), 345–379 (1981). https://doi.org/10.1002/prop.19810290802

    Article  ADS  MathSciNet  Google Scholar 

  16. K. Nagata, There is no axiomatic system for the quantum theory. Int. J. Theor. Phys. 48, 3532 (2009). https://doi.org/10.1007/s10773-009-0158-z

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Nagata, T. Nakamura, Can von Neumann’s Theory meet the Deutsch–Jozsa algorithm? Int. J. Theor. Phys. 49, 162 (2010). https://doi.org/10.1007/s10773-009-0189-5

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Nagata, D. Ngoc Diep, T. Nakamura, Two symmetric measurements may cause an unforeseen effect. Quant. Inform. Process. 22(2), 94 (2023). https://doi.org/10.1007/s11128-023-03841-5

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. J.A. de Barros, Comments on “There is no axiomatic system for the quantum theory". Int. J. Theor. Phys. 50, 1828 (2011). https://doi.org/10.1007/s10773-011-0696-z

    Article  MATH  Google Scholar 

  20. A. Avron, A. Zamansky, Non-Deterministic Semantics for Logical Systems, in Handbook of Philosophical Logic, vol. 16, ed. by D. Gabbay, F. Guenthner (Springer, Dordrecht, 2011). https://doi.org/10.1007/978-94-007-0479-4_4

    Chapter  Google Scholar 

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Acknowledgements

We thank Soliman Abdalla, Jaewook Ahn, Josep Batle, Do Ngoc Diep, Mark Behzad Doost, Ahmed Farouk, Han Geurdes, Preston Guynn, Shahrokh Heidari, Wenliang Jin, Hamed Daei Kasmaei, Janusz Milek, Mosayeb Naseri, Santanu Kumar Patro, Germano Resconi, and Renata Wong for their valuable support.

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Authors and Affiliations

  1. Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea

    Koji Nagata

  2. Department of Information and Computer Science, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Tadao Nakamura

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  1. Koji Nagata
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  2. Tadao Nakamura
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Koji Nagata and Tadao Nakamura wrote and read the manuscript.

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Correspondence to Koji Nagata.

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Nagata, K., Nakamura, T. How quantum mechanics might be when measuring commuting observables. Appl. Phys. A 129, 760 (2023). https://doi.org/10.1007/s00339-023-07043-9

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