  We seek to develop a Bures (minimal monotone/statistical distinguishability)
metric-based series of formulas for the moments of probability distributions
over the determinants $|\rho|$ and $|\rho^{PT}|$ of $4 \times 4$ density
matrices, $\rho$, for generalized (rebit, quater[nionic]bit,\ldots) two-qubit
systems, analogous to a series that has been obtained for the Hilbert-Schmidt
(HS) metric. In particular, we desire--using moment-inversion procedures--to be
able to closely test the previously-developed conjecture (J. Geom. Phys., 53,
74 [2005]) that the Bures separability probability over the (standard,
fifteen-dimensional convex set of) two-qubit states is $\frac{1680
\left(\sqrt{2}-1\right)}{\pi ^8} \approx 0.0733389$--while, in the HS context,
strong evidence has been adduced, along the indicated analytical lines, that
the counterpart of this value is $\frac{8}{33}$ (J. Phys. A}, 45, 095305
[2012]). Working within the "utility function" framework of Dunkl employed in
that latter study, we obtain an interesting $_{10}F_{9}$ balanced
hypergeometric function based on a "hybridization" of known Bures and HS terms.
This exercise appears to provide an upper bound on the Bures two-qubit
separability probability of 0.0798218. We also examine the yet unresolved HS
qubit-qutrit scenario. Mathematica calculations indicate that if the same form
of hypergeometric paradigm as has been established for the generalized
two-qubit HS moments is followed in either the HS qubit-qutrit or Bures
two-qubit cases, then a balanced hypergeometric function $_{p}F_{p-1}$ with
$p>9$ would be required.
