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The nonnegativity of the determinant of the partial transpose of a two-qubit(4 x 4) density matrix is both a necessary and sufficient condition for itsseparability. While the determinant is restricted to the interval [0,1/256],the determinant of the partial transpose can range over [-1/16,1/256], withnegative values corresponding to entangled states. We report here the exactvalues of the first nine moments of the probability distribution of the partialtranspose over this interval, with respect to the Hilbert-Schmidt (metricvolume element) measure on the nine-dimensional convex set of real two-qubitdensity matrices. Rational functions C_{2 j}(m), yielding the coefficients ofthe 2j-th power of even polynomials occurring at intermediate steps in ourderivation of the m-th moment, emerge. These functions possess poles at finiteseries of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain(trivial) roots at finite series of consecutive natural numbers (m=0, 1,...).Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the samehalf-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The firsttwo moments (mean and variance) found--when employed in the one-sided Chebyshevinequality--give an upper bound of 30397/34749 = 0.874759 on the separabilityprobability of real two-qubit density matrices. We are able to report generalformulas for the m-th moment of the Hilbert-Schmidt probability distribution ofthe density matrix determinant over [0,1/256], in the real, complex andquaternionic two-qubit cases.

Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes