Von Neumann Normalisation and Symptoms of Randomness: An Application to Sequences of Quantum Random Bits

  • Alastair A. Abbott
  • Cristian S. Calude
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6714)


Due to imperfections in measurement and hardware, the flow of bits generated by a quantum random number generator (QRNG) contains bias and correlation, two symptoms of non-randomness. There is no algorithmic method to eliminate correlation as this amounts to guaranteeing incomputability. However, bias can be mitigated: QRNGs use normalisation techniques such as von Neumann’s method—the first and simplest technique for reducing bias—and other more efficient modifications.

In this paper we study von Neumann un-biasing normalisation for an ideal QRNG operating ‘to infinity’, i.e. producing an infinite bit-sequence. We show that, surprisingly, von Neumann un-biasing normalisation can both increase or decrease the (algorithmic) randomness of the generated sequences. The impact this has on the quality of incomputability of sequences of bits from QRNGs is discussed.

A successful application of von Neumann normalisation—in fact, any un-biasing transformation—does exactly what it promises, un-biasing, one (among infinitely many) symptoms of randomness; it will not produce ‘true’ randomness, a mathematically vacuous concept.


Probability Space Turing Machine Shot Noise Quantum Randomness Probabilistic Treatment 
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  1. 1.
    Abbott, A.A., Calude, C.S.: Von Neumann normalisation of a quantum random number generator. Report CDMTCS-392, Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland, Auckland, New Zealand (2010)Google Scholar
  2. 2.
    Abbott, A.A., Calude, C.S., Svozil, K.: Unpublished work on the incomputability of quantum randomness (in preparation)Google Scholar
  3. 3.
    Abbott, A.A., Calude, C.S., Svozil, K.: A quantum random number generator certified by value indefiniteness. CDMTCS Research Report, 396 (2010)Google Scholar
  4. 4.
    Bell, J.S.: On the problem of hidden variables in quantum mechanics. Reviews of Modern Physics 38(3), 447–452 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Billingsley, P.: Probability and Measure. John Wiley & Sons, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Born, M.: Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 38, 803–837 (1926); English translation by Wheeler, J. A., Zurek, W.H.: Quantum Theory and Measurement, ch. I.2. Princeton University Press, Princeton (1983) CrossRefzbMATHGoogle Scholar
  7. 7.
    Calude, C.S.: Information and Randomness: An Algorithmic Perspective, 2nd edn. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Calude, C.S., Hertling, P., Jürgensen, H., Weihrauch, K.: Randomness on full shift spaces. Chaos, Solutions & Fractals 12(3), 491–503 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Calude, C.S., Svozil, K.: Quantum randomness and value indefiniteness. Advanced Science Letters 1, 165–168 (2008)CrossRefGoogle Scholar
  10. 10.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Halmos, P.R.: Measure Theory. Springer, New York (1974)zbMATHGoogle Scholar
  12. 12.
    Heywood, P., Redhead, M.L.G.: Nonlocality and the Kochen-Specker paradox. Foundations of Physics 13(5), 481–499 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    id Quantique. Quantis—quantum random number generators (12/08/2009),
  14. 14.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading (1968)zbMATHGoogle Scholar
  15. 15.
    Jennewein, T., Achleitner, U., Weihs, G., Weinfurter, H., Zeilinger, A.: A fast and compact quantum random number generator. Review of Scientific Instruments 71, 1675–1680 (2000)CrossRefGoogle Scholar
  16. 16.
    Kac, M.: Statistical Independence in Probability, Analysis and Number Theory. The Carus Mathematical Monographs. The Mathematical Association of America (1959)Google Scholar
  17. 17.
    Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 17, 59–87 (1967); Reprinted in Specker, E.: Selecta. Brikhäuser, Basel (1990)Google Scholar
  18. 18.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. John Wiley & Sons, New York (1974)zbMATHGoogle Scholar
  19. 19.
    Kwon, O., Cho, Y., Kim, Y.: Quantum random number generator using photon-number path entanglement. Applied Optics 48(9), 1774–1778 (2009)CrossRefGoogle Scholar
  20. 20.
    Ma, H., Wang, S., Zhang, D., Change, J., Ji, L., Hou, Y., Wu, L.: A random-number generator based on quantum entangled photon pairs. Chinese Physics Letters 21(19), 1961–1964 (2004)Google Scholar
  21. 21.
    Martin-Löf, P.: The definition of random sequences. Information and Control 9(6), 602–619 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Peres, Y.: Iterating von Neumann’s procedure for extracting random bits. The Annals of Statistics 20(1), 590–597 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pironio, S., Acín, A., Massar, S., de la Giroday, A.B., Matsukevich, D.N., Maunz, P., Olmchenk, S., Hayes, D., Luo, L., Manning, T.A., Monroe, C.: Random numbers certified by Bell’s theorem. Nature 464(09008) (2010)Google Scholar
  24. 24.
    Schmidt, H.: Quantum-mechanical random-number generator. Journal of Applied Physics 41(2), 462–468 (1970)CrossRefGoogle Scholar
  25. 25.
    Shen, Y., Tian, L., Zou, H.: Practical quantum random number generator based on measuring the shot noise of vacuum states. Physical Review A 81(063814) (2010)Google Scholar
  26. 26.
    Stefanov, A., Gisin, N., Guinnard, O., Guinnard, L., Zbinden, H.: Optical quantum random number generator. Journal of Modern Optics 47(4), 595–598 (2000)Google Scholar
  27. 27.
    Svozil, K.: The quantum coin toss – testing microphysical undecidability. Physics Letters A 143(9), 433–437 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Svozil, K.: Quantum information via state partitions and the context translation principle. Journal of Modern Optics 51, 811–819 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Svozil, K.: Three criteria for quantum random-number generators based on beam splitteres. Physical Review A 79(5), 054306 (2009)CrossRefGoogle Scholar
  30. 30.
    Vadhan, S.: Pseudorandomness. Foundations and Trends in Theoretical Computer Science. Now publishers (to appear, 2011)Google Scholar
  31. 31.
    von Neumann, J.: Various techniques used in connection with random digits. National Bureau of Standards Applied Math Series 12, 36–38 (1951); In: Traub, A.H. (ed.) John von Neumann, Collected Works, pp. 768–770. MacMillan, New York (1963) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alastair A. Abbott
    • 1
  • Cristian S. Calude
    • 1
  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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