Foundations of Physics

, Volume 41, Issue 3, pp 345–356 | Cite as

A Quantum-Bayesian Route to Quantum-State Space

  • Christopher A. Fuchs
  • Rüdiger SchackEmail author


In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent’s personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor.


Quantum foundations Quantum state space Quantum Bayesianism Born rule Bayesian 


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  1. 1.
    Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43, 4537 (2002) zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Fuchs, C.A.: Notes on a Paulian Idea: Foundational, Historical, Anecdotal & Forward-Looking Thoughts on the Quantum. Växjö University Press, Växjö (2003). With foreword by N. David Mermin. Preprinted as arXiv:quant-ph/0105039v1 (2001) Google Scholar
  3. 3.
    Schack, R., Brun, T.A., Caves, C.M.: Quantum Bayes rule. Phys. Rev. A 64, 014305 (2001) CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Fuchs, C.A.: Quantum mechanics as quantum information (and only a little more). arXiv:quant-ph/0205039v1 (2002); abridged version in: Khrennikov, A. (ed.) Quantum Theory: Reconsideration of Foundations, pp. 463–543. Växjö University Press, Växjö (2002)
  5. 5.
    Fuchs, C.A.: Quantum mechanics as quantum information, mostly. J. Mod. Opt. 50, 987 (2003) zbMATHADSGoogle Scholar
  6. 6.
    Schack, R.: Quantum theory from four of Hardy’s axioms. Found. Phys. 33, 1461 (2003) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fuchs, C.A., Schack, R.: Unknown quantum states and operations, a Bayesian view. In: Paris, M.G.A., Řeháček, J. (eds.) Quantum Estimation Theory, pp. 151–190. Springer, Berlin (2004) Google Scholar
  8. 8.
    Caves, C.M., Fuchs, C.A., Schack, R.: Subjective probability and quantum certainty. Stud. Hist. Philos. Mod. Phys. 38, 255 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Appleby, D.M.: Facts, values and quanta. Found. Phys. 35, 627 (2005) zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Appleby, D.M.: Probabilities are single-case, or nothing. Opt. Spectrosc. 99, 447 (2005) CrossRefADSGoogle Scholar
  11. 11.
    Timpson, C.J.: Quantum Bayesianism: a study. Stud. Hist. Philos. Mod. Phys. 39, 579 (2008) CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Fuchs, C.A., Schack, R.: Quantum-Bayesian coherence. Rev. Mod. Phys. (2009, submitted). arXiv:0906.2187v1 [quant-ph]
  13. 13.
    Ramsey, F.P.: Truth and probability. In: Braithwaite, R.B. (ed.) The Foundations of Mathematics and Other Logical Essays, pp. 156–198. Harcourt Brace, New York (1931) Google Scholar
  14. 14.
    de Finetti, B.: Probabilismo. Logos 14, 163 (1931); transl., Probabilism. Erkenntnis 31, 169 (1989) Google Scholar
  15. 15.
    Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954) zbMATHGoogle Scholar
  16. 16.
    de Finetti, B.: Theory of Probability. Wiley, New York (1990), 2 volumes zbMATHGoogle Scholar
  17. 17.
    Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, Chichester (1994) zbMATHCrossRefGoogle Scholar
  18. 18.
    Jeffrey, R.: Subjective Probability. The Real Thing. Cambridge University Press, Cambridge (2004) zbMATHGoogle Scholar
  19. 19.
    Logue, J.: Projective Probability. Oxford University Press, Oxford (1995) Google Scholar
  20. 20.
    Appleby, D.M., Flammia, S.T., Fuchs, C.A.: The Lie algebraic significance of symmetric informationally complete measurements. J. Math. Phys. (2010, submitted). arXiv:1001.0004v1 [quant-ph]
  21. 21.
    Appleby, D.M., Ericsson, Å., Fuchs, C.A.: Pseudo-QBist State Spaces. Found. Phys. (2009, accepted) Google Scholar
  22. 22.
    Skyrms, B.: Coherence. In: Rescher, N. (ed.) Scientific Inquiry in Philosophical Perspective, pp. 225–242. University of Pittsburgh Press, Pittsburgh (1987) Google Scholar
  23. 23.
    Zauner, G.: Quantum designs—foundations of a non-commutative theory of designs (in German). PhD thesis, University of Vienna (1999) Google Scholar
  24. 24.
    Caves, C.M.: Symmetric informationally complete POVMs. Unpublished (1999) Google Scholar
  25. 25.
    Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Fuchs, C.A.: On the quantumness of a Hilbert space. Quantum. Inf. Comput. 4, 467 (2004) zbMATHMathSciNetGoogle Scholar
  27. 27.
    Appleby, D.M.: SIC-POVMs and the extended Clifford group. J. Math. Phys. 46, 052107 (2005) CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Appleby, D.M., Dang, H.B., Fuchs, C.A.: Physical significance of symmetric informationally-complete sets of quantum states. arXiv:0707.2071v1 [quant-ph] (2007)
  29. 29.
    Scott, A.J., Grassl, M.: SIC-POVMs: a new computer study. arXiv:0910.5784v2 [quant-ph] (2009)
  30. 30.
    Ferrie, C., Emerson, J.: Framed Hilbert space: hanging the quasi-probability pictures of quantum theory. New J. Phys. 11, 063040 (2009) CrossRefADSGoogle Scholar
  31. 31.
    Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys. 16, 391 (1986) CrossRefMathSciNetADSGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of Mathematics, Royal HollowayUniversity of LondonEgham, SurreyUK

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