  We extend to additional probability measures and scenarios, certain of the
recent results of Krattenthaler and Slater (quant-ph/9612043), whose original
motivation was to obtain quantum analogs of seminal work on universal data
compression of Clarke and Barron. KS obtained explicit formulas for the
eigenvalues and eigenvectors of the 2^m x 2^m density matrices derived by
averaging the m-fold tensor products with themselves of the 2 x 2 density
matrices. The weighting was done with respect to a one-parameter (u) family of
probability distributions, all the members of which are spherically-symmetric
(SU(2)-invariant) over the ``Bloch sphere'' of two-level quantum systems. For u
= 1/2, one obtains the normalized volume element of the minimal monotone
(Bures) metric. In this paper, analyses parallel to those of KS are conducted,
based on an alternative "natural" measure on the density matrices recently
proposed by Zyczkowski, Horodecki, Sanpera, and Lewenstein (quant-ph/9804024).
The approaches of KS and that based on ZHSL are found to yield [1 + m/2]
identical SU(2) x S_{m}-invariant eigenspaces (but not coincident eigenvalues
for m > 3). Companion results, based on the SU(3) form of the ZHSL measure, are
obtained for the twofold and threefold tensor products of the 3 x 3 density
matrices. We find a rather remarkable limiting procedure (selection rule) for
recovering from these analyses, the (permutationally-symmetrized) multiplets of
SU(3) constructed from two or three particles. We also analyze the scenarios
(all for m = 2) N = 2 x 3, N= 2 x 3 x 2, N= 3 x 2 x 2 and N = 4 and, in
addition, generalize the ZHSL measure, so that it incorporates a family of
(symmetric) Dirichlet distributions (rather than simply the uniform
distribution), defined on the (N-1)-dimensional simplex of eigenvalues.
