Von Neumann normalisation and symptoms of randomness: An application to sequences of quantum random bits

Abstract

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. © 2011 Springer-Verlag Berlin Heidelberg.