International Journal of Theoretical Physics

, Volume 56, Issue 12, pp 3727–3739 | Cite as

The Extended Bloch Representation of Entanglement and Measurement in Quantum Mechanics

  • Diederik Aerts
  • Massimiliano Sassoli de Bianchi
  • Sandro Sozzo


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.


Measurement problem Hidden-variables Hidden-measurements Bloch sphere Extended Bloch representation SU(NEntanglement 


  1. 1.
    Busch, P., Lahti, P.J., Mittelstaedt, P.: The Quantum Theory of Measurement. Springer, Berlin (1991)CrossRefzbMATHGoogle 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)ADSCrossRefGoogle Scholar
  3. 3.
    van Fraassen, B.C.: Quantum mechanics: an empiricist view. Oxford University Press, Oxford (1991)Google Scholar
  4. 4.
    Aerts, D.: The description of joint quantum entities and the formulation of a paradox. Int. J. Theor. Phys. 39, 485–496 (2000)zbMATHMathSciNetGoogle 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)ADSCrossRefMathSciNetGoogle 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)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Beltrametti, E.G., Cassinelli, G.: The logic of quantum mechanics. Addison-Wesley, Reading (1981)zbMATHGoogle 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)ADSCrossRefGoogle Scholar
  10. 10.
    Kastner, R.E.: The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility. Cambridge University Press, New York (2013)zbMATHGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kimura, G.: The Bloch vector for N-level systems. Phys. Lett. A 314, 339 (2003)ADSCrossRefzbMATHMathSciNetGoogle 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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Aerts, D., Sassoli de Bianchi, M.: Many-measurements or many-worlds? a dialogue. Found. Sci. 20, 399–427 (2015)CrossRefMathSciNetGoogle 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)ADSCrossRefMathSciNetGoogle 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)Google Scholar
  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)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Diederik Aerts
    • 1
  • Massimiliano Sassoli de Bianchi
    • 2
  • Sandro Sozzo
    • 3
  1. 1.Center Leo Apostel (Clea)Brussels Free University (VUB)BrusselBelgium
  2. 2.Laboratorio di Autoricerca di BaseLuganoSwitzerland
  3. 3.School of Business and Institute IQSCSUniversity of LeicesterLeicesterUK

Personalised recommendations