Skip to main content

Information Invariance and Quantum Probabilities

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.

This is a preview of subscription content, access via your institution.

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)

    MATH  Article  ADS  Google 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)

    Article  ADS  MathSciNet  Google 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)

    Article  ADS  MathSciNet  Google Scholar 

  7. 7.

    Leggett, A.J.: Nonlocal hidden-variable theories and quantum mechanics: an incompatibility theorem. Found. Phys. 33, 1469 (2003)

    Article  MathSciNet  Google 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)

    Article  ADS  Google 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)

    Article  ADS  Google Scholar 

  10. 10.

    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24, 379 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  11. 11.

    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75, 032304 (2007)

    Article  ADS  Google 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)

    Article  MathSciNet  Google Scholar 

  15. 15.

    Hardy, L.: Quantum ontological excess baggage. Stud. Hist. Philos. Mod. Phys. 35, 267 (2004)

    Article  MathSciNet  Google Scholar 

  16. 16.

    Montina, A.: Exponential growth of the ontological space dimension with the physical size. Phys. Rev. A 77, 022104 (2008)

    Article  ADS  Google 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)

    Article  ADS  Google Scholar 

  18. 18.

    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696 (1935)

    MATH  Article  ADS  Google Scholar 

  19. 19.

    Wootters, W.K.: Statistical distance and Hilbert space. Phys. Rev. D 23, 357 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  20. 20.

    Fivel, D.I.: How interference effects in mixtures determine the rules of quantum mechanics. Phys. Rev. A 59, 2108 (1994)

    Article  ADS  Google Scholar 

  21. 21.

    Summhammer, J.: Maximum predictive power and the superposition principle. Int. J. Theor. Phys. 33, 171 (1994)

    Article  MathSciNet  Google 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)

    Article  ADS  MathSciNet  Google Scholar 

  24. 24.

    Caticha, A.: Consistency, amplitudes and probabilities in quantum theory. Phys. Rev. A 57, 1572 (1998)

    Article  ADS  Google 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)

    Article  MathSciNet  Google Scholar 

  30. 30.

    Grangier, P.: Contextual objectivity: a realistic interpretation of quantum mechanics. Eur. J. Phys. 23, 331 (2002). arXiv:quant-ph/0012122

    Article  Google 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

  32. 32.

    Luo, S.: Maximum Shannon entropy, minimum Fisher information, and an elementary game. Found. Phys. 32, 1757 (2002)

    Article  MathSciNet  Google 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)

    MATH  Article  Google Scholar 

  34. 34.

    Spekkens, R.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75, 032110 (2007)

    Article  ADS  Google Scholar 

  35. 35.

    Goyal, P.: Information-geometric reconstruction of quantum theory. Phys. Rev. A 78, 052120 (2008)

    Article  ADS  Google 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)

    Article  ADS  MathSciNet  Google Scholar 

  40. 40.

    Zyczkowski, K.: Quartic quantum theory: an extension of the standard quantum mechanics. J. Phys. A 41, 355302–23 (2008)

    Article  MathSciNet  Google 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)

    Article  ADS  MathSciNet  Google Scholar 

  43. 43.

    Ivanovic, I.: Geometrical description of quantal state determination. J. Phys. A 14, 3241 (1981)

    Article  ADS  MathSciNet  Google 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)

    Article  ADS  Google Scholar 

  45. 45.

    Greenberger, D., Yasin, A.: Simultaneous wave and particle knowledge in a neutron interferometer. Phys. Lett. A 128, 391 (1988)

    Article  ADS  Google Scholar 

  46. 46.

    Jaeger, G., Shimony, A., Vaidman, L.: Two interferometric complementarities. Phys. Rev. A 51, 54–67 (1995)

    Article  ADS  Google Scholar 

  47. 47.

    Englert, B.G.: Fringe visibility and which-way information: an inequality. Phys. Rev. Lett. 77, 2154 (1996)

    Article  ADS  Google Scholar 

  48. 48.

    Brukner, Č., Zeilinger, A.: Operationally invariant information in quantum measurements. Phys. Rev. Lett. 83, 3354–3357 (1999)

    MATH  Article  ADS  MathSciNet  Google 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)

    MATH  MathSciNet  Google Scholar 

  51. 51.

    Tsallis, C.: Possible generalizations of the Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  52. 52.

    Gisin, N.: Weinberg’s non-linear quantum mechanics and superluminal communications. Phys. Lett. A 143, 1–2 (1990)

    Article  ADS  Google Scholar 

  53. 53.

    Peres, A.: Nonlinear variants of Schrödinger’s equation violate the second law of thermodynamics. Phys. Rev. Lett. 63, 1114–1114 (1989)

    Article  ADS  Google 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)

    Article  ADS  Google Scholar 

  55. 55.

    Hardy, L.: Probability theories in general and quantum theory in particular. Stud. Hist. Philos. Mod. Phys. 34, 381–393 (2003)

    Article  MathSciNet  Google Scholar 

  56. 56.

    Brukner, Č, Zeilinger, A.: Conceptual inadequacy of the Shannon information in quantum measurements. Phys. Rev. A 63, 022113 (2001)

    Article  ADS  Google Scholar 

  57. 57.

    Nha, H., Zubairy, M.S.: Uncertainty inequalities as entanglement criteria for negative partial-transpose states. Phys. Rev. Lett. 101, 130402 (2008)

    Article  ADS  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Časlav Brukner.

Additional information

This article is dedicated to Pekka Lahti on the occasion of his 60th birthday.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brukner, Č., Zeilinger, A. Information Invariance and Quantum Probabilities. Found Phys 39, 677–689 (2009). https://doi.org/10.1007/s10701-009-9316-7

Download citation

Keywords

  • Quantum theory
  • Information
  • Measures of information
  • Alpha-entropy
  • Probabilistic theory