  A complete description of the multitudinous ways in which quantum particles
can be entangled requires the use of high-dimensional abstract mathematical
spaces. We report here a particularly interesting feature of the
nine-dimensional convex set-endowed with Hilbert-Schmidt (Euclidean/flat)
measure-composed of two-rebit (4 x 4) density matrices (rho). To each rho is
assigned the product of its (nonnegative) determinant |rho| and the determinant
of its partial transpose |rho^{PT}| -negative values of which, by the results
of Peres and Horodecki, signify the entanglement of rho. Integrating this
product, |rho| |rho^{PT}| =|rho rho^{PT}|, over the nine-dimensional space,
using the indicated (HS) measure, we obtain the result zero. The two
determinants, thus, form a pair of multivariate orthogonal polynomials with
respect to HS measure. It is hypothesized that the analogous two determinants
for the standard fifteen-dimensional convex set of two-qubit density matrices
are similarly orthogonal. However, orthogonality does not hold, we find, if the
symmetry of the nine-dimensional two-rebit scenario is broken slightly, nor
with the use of non-flat measures, such as the prominent Bures (minimal
monotone) measure-nor in the full HS extension to the 15-dimensional convex set
of two-qubit density matrices. We discuss relations-involving the HS moments of
|\rho^{PT}|-to the long-standing problem of determining the probability that a
generic pair of rebits/qubits is separable.
