  In this paper we study von Neumann un-biasing normalisation for ideal and
real quantum random number generators, operating on finite strings or infinite
bit sequences. In the ideal cases one can obtain the desired un-biasing. This
relies critically on the independence of the source, a notion we rigorously
define for our model. In real cases, affected by imperfections in measurement
and hardware, one cannot achieve a true un-biasing, but, if the bias "drifts
sufficiently slowly", the result can be arbitrarily close to un-biasing. For
infinite sequences, normalisation can both increase or decrease the
(algorithmic) randomness of the generated sequences. 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.
