Foundations of Physics

, Volume 47, Issue 7, pp 991–1002 | Cite as

A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics

  • Alessio Benavoli
  • Alessandro Facchini
  • Marco Zaffalon
Article

Abstract

Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for \(n=2\). The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration.

Keywords

Gleason’s type theorem Dispersion-free probabilities Desirable gambles 

References

  1. 1.
    Anscombe, F.J., Aumann, R.J.: A definition of subjective probability. Ann. Math. Stat. 34, 199–2005 (1963)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barnett, S.M., Cresser, J.D., Jeffers, J., Pegg, D.T.: Quantum probability rule: a generalization of the theorems of Gleason and Busch. New J. Phys. 16(4), 043025 (2014)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38(3), 447 (1966)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benavoli, A., Facchini, A., Zaffalon, M.: Quantum mechanics: the Bayesian theory generalised to the space of Hermitian matrices. Phys. Rev. A. arXiv:1605.08177 (2016)
  5. 5.
    Busch, P.: Quantum states and generalized observables: a simple proof of Gleason’s theorem. Phys. Rev. Lett. 91(12), 120403 (2003)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43(9), 4537–4559 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Caves, C.M., Fuchs, C.A., Manne, K.K., Renes, A.M.: Gleason-type derivations of the quantum probability rule for generalized measurements. Found. Phys. 34(2), 193–209 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    de Finetti, B.: La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré 7, 1–68 (1937), English translation in Kyburg Jr. and Smokler (1964)Google Scholar
  9. 9.
    Fuchs, C.A., Schack, R.: A quantum-Bayesian route to quantum-state space. Found. Phys. 41(3), 345–356 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fuchs, C.A., Schack, R.: Quantum-Bayesian coherence. Rev. Mod. Phys. 85(4), 1693 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    Galaabaatar, T., Karni, E.: Subjective expected utility with incomplete preferences. Econometrica 81(1), 255–284 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6(6), 885–893 (1957)MathSciNetMATHGoogle Scholar
  13. 13.
    Heinosaari, T., Ziman, M.: The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement. Cambridge University Press, Cambridge (2011)CrossRefMATHGoogle Scholar
  14. 14.
    Holevo, A.S.: Statistical decision theory for quantum systems. J. Multivar. Anal. 3(4), 337–394 (1973)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982)MATHGoogle Scholar
  16. 16.
    Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)MathSciNetMATHGoogle Scholar
  17. 17.
    Kyburg, H.E. Jr., Smokler, H.E. (eds): Studies in Subjective Probability. Wiley, New York, second edition (with new material 1980) (1964)Google Scholar
  18. 18.
    Mermin, N.D.: Physics: QBism puts the scientist back into science. Nature 507(7493), 421–423 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    Pitowsky, I.: Betting on the outcomes of measurements: a Bayesian theory of quantum probability. Stud. Hist. Philos. Sci. Part B 34(3), 395–414 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pitowsky, I.: Quantum Mechanics as a Theory of Probability. In: Physical Theory and its Interpretation: Essays in Honor of Jeffrey Bub, pp 213–240. Springer, Netherlands (2006)Google Scholar
  21. 21.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)CrossRefMATHGoogle Scholar
  22. 22.
    Williams, P.M.: Notes on conditional previsions. Tech. rep., School of Mathematical and Physical Science, University of Sussex, UK, reprinted in Williams (2007) (1975)Google Scholar
  23. 23.
    Williams, P.M.: Notes on conditional previsions. Int. J. Approx. Reason. 44, 366–383, revised Journal version of Williams (1975) (2007)Google Scholar
  24. 24.
    Zaffalon, M., Miranda, E.: Desirability and the birth of incomplete preferences. Tech. Rep. abs/1506.00529, CoRR, downloadable at http://arxiv.org/abs/1506.00529 (2015)

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA)LuganoSwitzerland

Personalised recommendations