  We investigate the question of whether or not there exists a noncommutative/
quantum extension of a recent (commutative probabilistic) result of Clarke and
Barron. They demonstrated that the Jeffreys' invariant prior of Bayesian theory
yields the common asymptotic (minimax and maximin) redundancy - the excess of
the encoding cost over the source entropy - of universal data compression in a
parametric setting. We study certain probability distributions for the
two-level quantum systems. We are able to compute exact formulas for the
corresponding redundancies, for which we find the asymptotic limits. These
results are very suggestive and do indeed point towards a possible quantum
extension of the result of Clarke and Barron.
