The Extended Bloch Representation of Entanglement and Measurement in Quantum Mechanics

Abstract

The quantum formalism can be completed by assuming that density operators also represent genuine states. An ‘extended Bloch representation’ (EBR) then results, in which not only the states but also the measurement-interactions can be described. Consequently, the Born rule can be obtained as an expression that quantifies the lack of knowledge about the measurement-interaction that is each time actualized, during a measurement. Entanglement can also be consistently described in the EBR, as it remains compatible with the principle according to which a composite entity exists only if its components also exist, and therefore are in well-defined states.

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References

  1. 1.

    Busch, P., Lahti, P.J., Mittelstaedt, P.: The Quantum Theory of Measurement. Springer, Berlin (1991)

    Google Scholar 

  2. 2.

    Schrœdinger, E.: Naturwissenschaftern 23, 807 (1935). English translation: Trimmer, J. D.: Proc. Am. Philos. Soc. 124, 323 (1980). Reprinted in: Wheeler, J. A., Zurek, W. H. (Eds.), Quantum Theory and Measurement. Princeton University Press, Princeton (1983)

    ADS  Article  Google Scholar 

  3. 3.

    van Fraassen, B.C.: Quantum mechanics: an empiricist view. Oxford University Press, Oxford (1991)

  4. 4.

    Aerts, D.: The description of joint quantum entities and the formulation of a paradox. Int. J. Theor. Phys. 39, 485–496 (2000)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Hughston, L.P., Jozsa, R., Wootters, W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A 183, 14–18 (1993)

    ADS  Article  MathSciNet  Google Scholar 

  6. 6.

    Beretta, G.P.: The Hatsopoulos-Gyftopoulos resolution of the Schroedinger-Park paradox about the concept of “state” in quantum statistical mechanics. Modern Phys. Lett. A 21, 2799–2811 (2006)

    ADS  Article  MathSciNet  Google Scholar 

  7. 7.

    Beltrametti, E.G., Cassinelli, G.: The logic of quantum mechanics. Addison-Wesley, Reading (1981)

    Google Scholar 

  8. 8.

    d’Espagnat, B.: Conceptual Foundations of Quantum Mechanics, 2nd Edn. Addison-Wesley, Reading (1976)

    Google Scholar 

  9. 9.

    Schlosshauer, M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267–1305 (2005)

    ADS  Article  Google Scholar 

  10. 10.

    Kastner, R.E.: The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility. Cambridge University Press, New York (2013)

    Google Scholar 

  11. 11.

    Aerts, D., Sassoli de Bianchi, M.: The unreasonable success of quantum probability I. Quantum measurements as uniform fluctuations. J. Math. Psychol. 67, 51–75 (2015a)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    De Zela, F.: Gleason-type theorem for projective measurements, including qubits: The Born rule beyond quantum physics. Found. Phys. (2016). doi:10.1007/s10701-016-0020-0

  13. 13.

    Hioe, F.T., Eberly, J.H.: N-level coherence vector and higher conservation laws in quantum optics and quantum mechanics. Phys. Rev. Lett. 47, 838–841 (1981)

    ADS  Article  MathSciNet  Google Scholar 

  14. 14.

    Kimura, G.: The Bloch vector for N-level systems. Phys. Lett. A 314, 339 (2003)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Aerts, D., Sassoli de Bianchi, M.: The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem. Ann. Phys. 351, 975–1025 (2014). Erratum: Ann. Phys. 366, 197–198 (2016)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Aerts, D., Sassoli de Bianchi, M.: Many-measurements or many-worlds? a dialogue. Found. Sci. 20, 399–427 (2015)

    Article  MathSciNet  Google Scholar 

  17. 17.

    Aerts, D., Sassoli de Bianchi, M.: The extended Bloch representation of quantum mechanics. Explaining superposition, interference and entanglement (2015). arXiv:1504.04781[quant-ph]

  18. 18.

    Gamel, O.: Entangled bloch spheres: Bloch matrix and Two-Qubit state space. Phys. Rev. A 93, 062320 (2016)

    ADS  Article  MathSciNet  Google Scholar 

  19. 19.

    Aerts, D., Sassoli de Bianchi, M. Aerts, D., De Ronde, C., Freytes, H., Giuntini, R (eds.): A Possible Solution to the Second Entanglement Paradox. World Scientific Publishing Company, Singapore (2016). in print)

  20. 20.

    Aerts, D.: A mechanistic classical laboratory situation violating the Bell inequalities with \(2\sqrt {2}\), exactly ‘in the same way’ as its violations by the EPR experiments. Helv. Phys. Acta 64, 1–23 (1991)

    MathSciNet  Google Scholar 

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Correspondence to Sandro Sozzo.

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Aerts, D., de Bianchi, M.S. & Sozzo, S. The Extended Bloch Representation of Entanglement and Measurement in Quantum Mechanics. Int J Theor Phys 56, 3727–3739 (2017). https://doi.org/10.1007/s10773-016-3257-7

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Keywords

  • Measurement problem
  • Hidden-variables
  • Hidden-measurements
  • Bloch sphere
  • Extended Bloch representation
  • SU(N)
  • Entanglement