  Clarke and Barron have recently shown that the Jeffreys' invariant prior of
Bayesian theory yields the common asymptotic (minimax and maximin) redundancy
of universal data compression in a parametric setting. We seek a possible
analogue of this result for the two-level {\it quantum} systems. We restrict
our considerations to prior probability distributions belonging to a certain
one-parameter family, $q(u)$, $-\infty < u < 1$. Within this setting, we are
able to compute exact redundancy formulas, for which we find the asymptotic
limits. We compare our quantum asymptotic redundancy formulas to those derived
by naively applying the classical counterparts of Clarke and Barron, and find
certain common features. Our results are based on formulas we obtain for the
eigenvalues and eigenvectors of $2^n \times 2^n$ (Bayesian density) matrices,
$\zeta_{n}(u)$. These matrices are the weighted averages (with respect to
$q(u)$) of all possible tensor products of $n$ identical $2 \times 2$ density
matrices, representing the two-level quantum systems. We propose a form of {\it
universal} coding for the situation in which the density matrix describing an
ensemble of quantum signal states is unknown. A sequence of $n$ signals would
be projected onto the dominant eigenspaces of $\ze_n(u)$.
