A Balanced ₁₀F₉ Hypergeometric Hybrid Hilbert-Schmidt/Bures Two-Qubit Function and Related Constructions

We seek to develop a Bures (minimal monotone/statistical distinguishability) metric-based series of formulas for the moments of probability distributions over the determinants |ρ| and |ρ^PT| of 4 times 4 density matrices, ρ, 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\(sqrt{2}-1right\)}{π⁸} 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 ₁₀F₉ 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 _pF_p-1 with p>9 would be required.