Information-Theoretic Postulates for Quantum Theory

  • Markus P. MüllerEmail author
  • Lluís MasanesEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 181)


Why are the laws of physics formulated in terms of complex Hilbert spaces? Are there natural and consistent modifications of quantum theory that could be tested experimentally? This book chapter gives a self-contained and accessible summary of our paper [New J. Phys. 13, 063001, 2011] addressing these questions, presenting the main ideas, but dropping many technical details. We show that the formalism of quantum theory can be reconstructed from four natural postulates, which do not refer to the mathematical formalism, but only to the information-theoretic content of the physical theory. Our starting point is to assume that there exist physical events (such as measurement outcomes) that happen probabilistically, yielding the mathematical framework of “convex state spaces”. Then, quantum theory can be reconstructed by assuming that (i) global states are determined by correlations between local measurements, (ii) systems that carry the same amount of information have equivalent state spaces, (iii) reversible time evolution can map every pure state to every other, and (iv) positivity of probabilities is the only restriction on the possible measurements.



Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. LM acknowledges support from CatalunyaCaixa.


  1. 1.
    S. Weinberg, Ann. Phys. NY 194, 336 (1989)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    N. Gisin, Weinberg’s non-linear quantum mechanics and supraluminal communications. Phys. Lett. A 143, 1–2 (1990)ADSCrossRefGoogle Scholar
  3. 3.
    C. Simon, V. Bužek, N. Gisin, No-signaling condition and quantum dynamics. Phys. Rev. Lett. 87, 170405 (2001)ADSCrossRefGoogle Scholar
  4. 4.
    G. Birkhoff, J. von Neumann, The logic of quantum mechanics. Ann. Math. 37, 823 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G.W. Mackey, The Mathematical Foundations of Quantum Mechanics (W. A. Benjamin Inc., New York, 1963)zbMATHGoogle Scholar
  6. 6.
    G. Ludwig, Foundations of Quantum Mechanics I and II (Springer, New York, 1985)CrossRefGoogle Scholar
  7. 7.
    E.M. Alfsen, F.W. Shultz, Geometry of State Spaces of Operator Algebras (Birkhäuser, Boston, 2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    L. Hardy, Quantum theory from five reasonable axioms, arxiv:quant-ph/0101012v4
  9. 9.
    B. Dakić, C. Brukner, Quantum theory and beyond: is entanglement special?, in Deep beauty, ed. by H. Halvorson (Cambridge Press, 2011), arXiv:0911.0695v1
  10. 10.
    Ll. Masanes, M.P. Müller, A derivation of quantum theory from physical requirements. New J. Phys. 13, 063001 (2011)Google Scholar
  11. 11.
    G. Chiribella, G.M. D’Ariano, P. Perinotti, Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011)ADSCrossRefGoogle Scholar
  12. 12.
    L. Hardy, Reformulating and Reconstructing Quantum Theory, arXiv:1104.2066v1
  13. 13.
    L. Hardy, The Operator Tensor Formulation of Quantum Theory. Phil. Trans. R. Soc. A 370, 3385–417 (2012), arXiv:1201.4390v1
  14. 14.
    M. Zaopo, Information Theoretic Axioms for Quantum Theory, arXiv:1205.2306
  15. 15.
    L. Hardy, Foliable Operational Structures for General Probabilistic Theories, in “Deep beauty”, Editor Hans Halvorson (Cambridge Press, 2011), arXiv:0912.4740v1
  16. 16.
    H. Barnum, A. Wilce, Information processing in convex operational theories, DCM/QPL (Oxford University 2008), arXiv:0908.2352v1
  17. 17.
    J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007), arXiv:quant-ph/0508211v3
  18. 18.
    G. Chiribella, G. M. D’Ariano, P. Perinotti; Probabilistic theories with purification; Phys. Rev. A 81, 062348 (2010), arXiv:0908.1583v5
  19. 19.
    A.S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    R. T. Rockafellar, Convex Analysis, (Princeton University Press, 1970)Google Scholar
  21. 21.
    G. Brassard, Is information the key? Nat. Phys. 1, 2 (2005)CrossRefGoogle Scholar
  22. 22.
    Č. Brukner, Questioning the rules of the game. Physics 4, 55 (2011)CrossRefGoogle Scholar
  23. 23.
    W.K. Wootters, Quantum mechanics without probability amplitudes. Found. Phys. 16, 391–405 (1986)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Baker, Matrix Groups, An Introduction to Lie Group Theory (Springer, London, 2006)Google Scholar
  25. 25.
    C.D. Aliprantis, R. Tourky, Cones and Duality, (American Mathematical Society, 2007)Google Scholar
  26. 26.
    S. Straszewicz, Über exponierte Punkte abgeschlossener Punktmengen. Fund. Math. 24, 139–143 (1935)zbMATHGoogle Scholar
  27. 27.
    M. Navascues, H. Wunderlich, A glance beyond the quantum model. Proc. R. Soc. Lond. A 466, 881–890 (2009). arXiv:0907.0372v1
  28. 28.
    W. van Dam, Implausible Consequences of Superstrong Nonlocality. Nat. Comput. 12(1), 9–12 (2012), arXiv:quant-ph/0501159v1
  29. 29.
    D. Gross, M. Müller, R. Colbeck, O.C.O. Dahlsten, All reversible dynamics in maximally non-local theories are trivial. Phys. Rev. Lett. 104, 080402 (2010), arXiv:0910.1840v2
  30. 30.
    M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, M. Zukowski, A new physical principle: information causality. Nature 461, 1101 (2009), arXiv:0905.2292v3
  31. 31.
    S. Popescu, D. Rohrlich, Causality and Nonlocality as Axioms for Quantum Mechanics, Proceedings of the Symposium on Causality and Locality in Modern Physics and Astronomy (York University, Toronto, 1997), arXiv:quant-ph/9709026v2
  32. 32.
    W. Fulton, J. Harris, Graduate texts in mathematics, Representation Theory (Springer, Berlin, 2004)Google Scholar
  33. 33.
    B. Simon, Representations of Finite and Compact Groups. Graduate Studies in Mathematics, vol. 10 (American Mathematical Society, Providence, 1996)Google Scholar
  34. 34.
    D. Montgomery, H. Samelson, Transformation groups of spheres. Ann. Math. 44, 454–470 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    A. Borel, Some remarks about Lie groups transitive on spheres and tori. Bull. A.M.S. 55, 580–587 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    A.L. Onishchik, V.V. Gorbatsevich, Lie Groups and Lie Algebras I, Encyclopedia of Mathematical Sciences 20 (Springer, Berlin, 1993)CrossRefGoogle Scholar
  37. 37.
    A. L. Onishchik, Transitive compact transformation groups, Mat. Sb. (N.S.) 60(102):4 447–485 (1963); English translation: Amer. Math. Soc. Transl. (2) 55, 153–194 (1966)Google Scholar
  38. 38.
    V. Bargmann, Note on Wigner’s theorem on symmetry operations. J. Math. Phys. 5, 862–868 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    E.P. Wigner, Normal form of antiunitary operators. J. Math. Phys. 1, 409–413 (1960)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    M.J. Bremner, C.M. Dawson, J.L. Dodd, A. Gilchrist, A.W. Harrow, D. Mortimer, M.A. Nielsen, T.J. Osborne, Practical scheme for quantum computation with any two-qubit entangling gate. Phys. Rev. Lett. 89, 247902 (2002), arXiv:quant-ph/0207072v1
  41. 41.
    T. Paterek, B. Dakić, Č. Brukner, Theories of systems with limited information content. New J. Phys. 12, 053037 (2010)ADSCrossRefGoogle Scholar
  42. 42.
    C. Ududec, H. Barnum, J. Emerson, Three slit experiments and the structure of quantum theory. Found. Phys. 41, 396–405 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ll. Masanes, M. P. Müller, D. Pérez-García, R. Augusiak, Entanglement and the three-dimensionality of the Bloch ball. J. Math. Phys. 55, 122203 (2014), arXiv:1111.4060
  44. 44.
    Ll. Masanes, M. P. Müller, R. Augusiak, D. Pérez-García, Existence of an information unit as a postulate of quantum theory. Proc. Natl. Acad. Sci. USA 110(41), 16373 (2013), arXiv:1208.0493
  45. 45.
    H. Barnum, A. Wilce, Local tomography and the Jordan structure of quantum theory. Found. Phys. 44, 192–212 (2014), arXiv:1202.4513
  46. 46.
    I. Bengtsson, K. Życzkowski, Geometry of Quantum States (University Press, Cambridge, 2006)CrossRefzbMATHGoogle Scholar
  47. 47.
    U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, G. Weihs, Ruling out multi-order interference in quantum mechanics. Science 329, 418–421 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Departments of Applied Mathematics and PhilosophyUniversity of Western OntarioLondonCanada
  2. 2.Department of Physics and AstronomyUniversity College LondonLondonUK

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