  We apply and extend recent results of Krattenthaler and Slater
(quant-ph/9612043), who sought 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 gotten 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 family of
probability distributions, all the members of which are spherically-symmetric
over the "Bloch sphere" of two-level quantum systems. This family includes the
normalized volume element of the minimal monotone (Bures) metric. In this
letter, we conduct parallel analyses (for m =2,3,4), based on a natural measure
on the density matrices recently proposed by Zyczkowski, Horodecki, Sanpera and
Lewenstein (quant-ph/9804024) and find interesting similarities and differences
with the findings of KS. In addition, we are able to obtain exact analogous
results, based on the measure of ZHSL, for the twofold tensor products of the 3
x 3 density matrices.
