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We seek to develop a Bures (minimal monotone/statistical distinguishability)metric-based series of formulas for the moments of probability distributionsover the determinants $|\rho|$ and $|\rho^{PT}|$ of $4 \times 4$ densitymatrices, $\rho$, for generalized (rebit, quater[nionic]bit,\ldots) two-qubitsystems, analogous to a series that has been obtained for the Hilbert-Schmidt(HS) metric. In particular, we desire--using moment-inversion procedures--to beable 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, thatthe counterpart of this value is $\frac{8}{33}$ (J. Phys. A}, 45, 095305[2012]). Working within the "utility function" framework of Dunkl employed inthat latter study, we obtain an interesting $_{10}F_{9}$ balancedhypergeometric function based on a "hybridization" of known Bures and HS terms.This exercise appears to provide an upper bound on the Bures two-qubitseparability probability of 0.0798218. We also examine the yet unresolved HSqubit-qutrit scenario. Mathematica calculations indicate that if the same formof hypergeometric paradigm as has been established for the generalizedtwo-qubit HS moments is followed in either the HS qubit-qutrit or Burestwo-qubit cases, then a balanced hypergeometric function $_{p}F_{p-1}$ with$p>9$ would be required.

A Balanced $_{10}F_{9}$ Hypergeometric Hybrid Hilbert-Schmidt/Bures Two-Qubit Function and Related Constructions