Foundations of Physics

, Volume 39, Issue 7, pp 677–689 | Cite as

Information Invariance and Quantum Probabilities

Article

Abstract

We consider probabilistic theories in which the most elementary system, a two-dimensional system, contains one bit of information. The bit is assumed to be contained in any complete set of mutually complementary measurements. The requirement of invariance of the information under a continuous change of the set of mutually complementary measurements uniquely singles out a measure of information, which is quadratic in probabilities. The assumption which gives the same scaling of the number of degrees of freedom with the dimension as in quantum theory follows essentially from the assumption that all physical states of a higher dimensional system are those and only those from which one can post-select physical states of two-dimensional systems. The requirement that no more than one bit of information (as quantified by the quadratic measure) is contained in all possible post-selected two-dimensional systems is equivalent to the positivity of density operator in quantum theory.

Keywords

Quantum theory Information Measures of information Alpha-entropy Probabilistic theory 

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References

  1. 1.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935) MATHCrossRefADSGoogle Scholar
  2. 2.
    Greenberger, D.M., Horne, M., Zeilinger, A.: Bell’s theorem. In: Kafatos, M. (ed.) Quantum Theory, and Conceptions of the Universe, pp. 73–76. Kluwer Academic, Dordrecht (1989) Google Scholar
  3. 3.
    Greenberger, D.M., Horne, M., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities Am. J. Phys. 58, 1131–1143 (1990) CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Bell, J.S.: Free variables and local causality. Dialectica 39, 103–106 (1985) Google Scholar
  5. 5.
    Bell, J.S.: On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200 (1964). Reprinted in J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univ. Press, Cambridge (1987) Google Scholar
  6. 6.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables, I and II. Phys. Rev. 85, 166–193 (1952) CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Leggett, A.J.: Nonlocal hidden-variable theories and quantum mechanics: an incompatibility theorem. Found. Phys. 33, 1469 (2003) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č, Zukowski, M., Aspelmeyer, M., Zeilinger, A.: An experimental test of non-local realism. Nature 446, 871–875 (2007) CrossRefADSGoogle Scholar
  9. 9.
    Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A., Scarani, V.: Experimental falsification of Leggett’s nonlocal variable model. Phys. Rev. Lett. 99, 210407 (2007) CrossRefADSGoogle Scholar
  10. 10.
    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24, 379 (1994) CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75, 032304 (2007) CrossRefADSGoogle Scholar
  12. 12.
    von Weizsäcker, C.F.: Aufbau der Physik. Hanser, Munich (1985). In German Google Scholar
  13. 13.
    Wheeler, J.A.: Information, physics, quantum: The search for links. In: Zurek, W. (ed.) Complexity, Entropy, and the Physics of Information. Addison-Wesley, Reading (1990) Google Scholar
  14. 14.
    Zeilinger, A.: A foundational principle for quantum mechanics. Found. Phys. 29, 631–643 (1999) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hardy, L.: Quantum ontological excess baggage. Stud. Hist. Philos. Mod. Phys. 35, 267 (2004) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Montina, A.: Exponential growth of the ontological space dimension with the physical size. Phys. Rev. A 77, 022104 (2008) CrossRefADSGoogle Scholar
  17. 17.
    Dakic, B., Suvakov, M., Paterek, T., Brukner, Č.: Efficient hidden-variable simulation of measurements in quantum experiments. Phys. Rev. Lett. 101, 190402 (2008) CrossRefADSGoogle Scholar
  18. 18.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696 (1935) MATHCrossRefADSGoogle Scholar
  19. 19.
    Wootters, W.K.: Statistical distance and Hilbert space. Phys. Rev. D 23, 357 (1981) CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Fivel, D.I.: How interference effects in mixtures determine the rules of quantum mechanics. Phys. Rev. A 59, 2108 (1994) CrossRefADSGoogle Scholar
  21. 21.
    Summhammer, J.: Maximum predictive power and the superposition principle. Int. J. Theor. Phys. 33, 171 (1994) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Summhammer, J.: Quantum theory as efficient representation of probabilistic information. arXiv:quant-ph/0701181
  23. 23.
    Bohr, A., Ulfbeck, O.: Primary manifestation of symmetry. Origin of quantal indeterminacy. Rev. Mod. Phys. 67, 1–35 (1995) CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Caticha, A.: Consistency, amplitudes and probabilities in quantum theory. Phys. Rev. A 57, 1572 (1998) CrossRefADSGoogle Scholar
  25. 25.
    Hardy, L.: Quantum theory from five reasonable axioms. arXiv.org/quant-ph/0101012.
  26. 26.
    Brukner, Č., Zeilinger, A.: Information and fundamental elements of the structure of quantum theory. In: Castell, L., Ischebeck, O. (ed.) Time, Quantum, Information. Springer, Berlin (2003). arXiv:quant-ph/0212084 Google Scholar
  27. 27.
    Brukner, Č., Zukowski, M.., Zeilinger, A.: The essence of entanglement. arXiv:quant-ph/0106119
  28. 28.
    Fuchs, C.: Quantum mechanics as quantum information (and only a little more). In: Khrenikov, A. (ed.) Quantum Theory: Reconstruction of Foundations. Växjo University Press, Växjo (2002). quant-ph/0205039 Google Scholar
  29. 29.
    Clifton, R., Bub, J., Halvorson, H.: Characterizing quantum theory in terms of information theoretic constraints. Found. Phys. 33, 1561 (2003) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Grangier, P.: Contextual objectivity: a realistic interpretation of quantum mechanics. Eur. J. Phys. 23, 331 (2002). arXiv:quant-ph/0012122 CrossRefGoogle Scholar
  31. 31.
    Grangier, P.: Contextual objectivity and the quantum formalism, Proc. of the conference “Foundations of Quantum Information” (April 2004, Camerino, Italy). quant-ph/0407025 Google Scholar
  32. 32.
    Luo, S.: Maximum Shannon entropy, minimum Fisher information, and an elementary game. Found. Phys. 32, 1757 (2002) CrossRefMathSciNetGoogle Scholar
  33. 33.
    Grinbaum, A.: Elements of information-theoretic derivation of the formalism of quantum theory. Int. J. Quant. Inf. 1(3), 289–300 (2003) MATHCrossRefGoogle Scholar
  34. 34.
    Spekkens, R.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75, 032110 (2007) CrossRefADSGoogle Scholar
  35. 35.
    Goyal, P.: Information-geometric reconstruction of quantum theory. Phys. Rev. A 78, 052120 (2008) CrossRefADSGoogle Scholar
  36. 36.
    D’Ariano, G.M.: Probabilistic theories: what is special about Quantum Mechanics? In: Philosophy of Quantum Information and Entanglement. Bokulich, A., Jaeger, G. (eds.), Cambridge University Press, Cambridge (to appear). arXiv:0807.4383
  37. 37.
    Dirac, P.A.M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1982) Google Scholar
  38. 38.
    Aaronson, S.: Is quantum mechanics an island in theoryspace? In: Khrennikov, A. (ed.). Proc. of the Växjö Conference “Quantum Theory: Reconsideration of Foundations”, 2004. quant-ph/0401062
  39. 39.
    Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys. 16, 391 (1986) CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Zyczkowski, K.: Quartic quantum theory: an extension of the standard quantum mechanics. J. Phys. A 41, 355302–23 (2008) CrossRefMathSciNetGoogle Scholar
  41. 41.
    Paterek, T., Dakic, B., Brukner, Č.: Theories of systems with limited information content. arXiv:0804.1423
  42. 42.
    Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363 (1989) CrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Ivanovic, I.: Geometrical description of quantal state determination. J. Phys. A 14, 3241 (1981) CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Wootters, W.K., Zurek, W.H.: Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle. Phys. Rev. D 19, 473 (1979) CrossRefADSGoogle Scholar
  45. 45.
    Greenberger, D., Yasin, A.: Simultaneous wave and particle knowledge in a neutron interferometer. Phys. Lett. A 128, 391 (1988) CrossRefADSGoogle Scholar
  46. 46.
    Jaeger, G., Shimony, A., Vaidman, L.: Two interferometric complementarities. Phys. Rev. A 51, 54–67 (1995) CrossRefADSGoogle Scholar
  47. 47.
    Englert, B.G.: Fringe visibility and which-way information: an inequality. Phys. Rev. Lett. 77, 2154 (1996) CrossRefADSGoogle Scholar
  48. 48.
    Brukner, Č., Zeilinger, A.: Operationally invariant information in quantum measurements. Phys. Rev. Lett. 83, 3354–3357 (1999) MATHCrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Bohr, N.: In: Atomic Physics and Human Knowledge. Wiley, New York (1958) Google Scholar
  50. 50.
    Havrda, J., Charvát, F.: Quantification method of classification processes. Concept of structural a-entropy. Kybernetika 3, 30–35 (1967) MATHMathSciNetGoogle Scholar
  51. 51.
    Tsallis, C.: Possible generalizations of the Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988) MATHCrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Gisin, N.: Weinberg’s non-linear quantum mechanics and superluminal communications. Phys. Lett. A 143, 1–2 (1990) CrossRefADSGoogle Scholar
  53. 53.
    Peres, A.: Nonlinear variants of Schrödinger’s equation violate the second law of thermodynamics. Phys. Rev. Lett. 63, 1114–1114 (1989) CrossRefADSGoogle Scholar
  54. 54.
    Abrams, D.S., Lloyd, S.: Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and P problems. Phys. Rev. Lett. 81, 3992–3995 (1998) CrossRefADSGoogle Scholar
  55. 55.
    Hardy, L.: Probability theories in general and quantum theory in particular. Stud. Hist. Philos. Mod. Phys. 34, 381–393 (2003) CrossRefMathSciNetGoogle Scholar
  56. 56.
    Brukner, Č, Zeilinger, A.: Conceptual inadequacy of the Shannon information in quantum measurements. Phys. Rev. A 63, 022113 (2001) CrossRefADSGoogle Scholar
  57. 57.
    Nha, H., Zubairy, M.S.: Uncertainty inequalities as entanglement criteria for negative partial-transpose states. Phys. Rev. Lett. 101, 130402 (2008) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute for Quantum Optics and Quantum InformationAustrian Academy of SciencesViennaAustria
  2. 2.Faculty of PhysicsUniversity of ViennaViennaAustria

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