A Balanced ${}_{10}{F}_9$ 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 $\mathrm{\mid }\rho \mathrm{\mid }$ and $\mathrm{\mid }{\rho }^{PT}\mathrm{\mid }$ 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.