Foundations of Physics

, Volume 42, Issue 3, pp 454–473 | Cite as

Limited Holism and Real-Vector-Space Quantum Theory

  • Lucien Hardy
  • William K. WoottersEmail author


Quantum theory has the property of “local tomography”: the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by “bilocal tomography”: the state of any composite system is determined by the statistics of measurements on pairs of components. Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that real-vector-space quantum theory, while not locally tomographic, is bilocally tomographic.


Holism Real-vector-space quantum theory Bilocal measurements State tomography General probabilistic theories 



Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.


  1. 1.
    Araki, H.: On a characterization of the state space of quantum mechanics. Commun. Math. Phys. 75, 1 (1980) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Bergia, S., Cannata, F., Cornia, A., Livi, R.: On the actual measurability of the density matrix of a decaying system by means of measurements on the decay products. Found. Phys. 10, 723 (1980) ADSCrossRefGoogle Scholar
  3. 3.
    Wootters, W.K.: Local accessibility of quantum states. In: Zurek, W.H. (ed.) Complexity, Entropy and the Physics of Information. Addison-Wesley, Reading (1990) Google Scholar
  4. 4.
    Mermin, N.D.: What is quantum mechanics trying to tell us? Am. J. Phys. 66, 753 (1998) ADSCrossRefGoogle Scholar
  5. 5.
    D’Ariano, G.M.: Candidates for principles of quantumness. (2009)
  6. 6.
    Stueckelberg, E.C.G.: Quantum theory in real Hilbert space. Helv. Phys. Acta 33, 727 (1960) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Mackey, G.: Mathematical Foundations of Quantum Mechanics. Benjamin, Elmsford (1963) zbMATHGoogle Scholar
  8. 8.
    Ludwig, G.: An Axiomatic Basis of Quantum Mechanics, Vols. 1 and 2. Springer-Verlag, Berlin (1985, 1987) CrossRefGoogle Scholar
  9. 9.
    Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970) MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys. 16, 391 (1986) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Foulis, D.J., Randall, C.H.: Empirical logic and tensor products. In: Neumann, H. (ed.) Interpretations and Foundations of Quantum Theory. Bibliographisches Institut, Wissenschaftsverlag, Mannheim (1981) Google Scholar
  12. 12.
    Gudder, S., Pulmannová, S., Bugajski, S., Beltrametti E.: Convex and linear effect algebras. Rep. Math. Phys. 44, 359–379 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Hardy, L.: Quantum theory from five reasonable axioms. (2001) arXiv:quant-ph/0101012
  14. 14.
    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75, 032304 (2007) ADSCrossRefGoogle Scholar
  15. 15.
    Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification (2009). arXiv:0908.1583
  16. 16.
    Hardy, L.: Foliable operational structures for general probabilistic theories (2009). arXiv:0912.4740
  17. 17.
    Hardy, L.: Reformulating and reconstructing quantum theory (2011). arXiv:1104.2066

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Perimeter InstituteWaterlooCanada
  2. 2.Department of PhysicsWilliams CollegeWilliamstownUSA
  3. 3.Department of Applied PhysicsKigali Institute of Science and TechnologyKigaliRwanda

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