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We seek to derive the probability--expressed in terms of the Hilbert-Schmidt(Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubitsystem is separable, by implementing the well-known Peres-Horodecki test on thepartial transposes (PTs) of the associated 4 x 4 density matrices. But the fullimplementation of the test--requiring that the determinant of the PT benonnegative for separability to hold--appears to be, at least presently,computationally intractable. So, we have previously implemented--using theauxiliary concept of a diagonal-entry-parameterized separability function(DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principalminors of the PT. This yielded an exact upper bound on the separabilityprobability of 1024/(135 Pi^2) = 0.76854. Here, we extend this line of work byrequiring that the four 3 x 3 principal minors of the PT be nonnegative, givingus an improved/reduced upper bound of 22/35 = 0.628571. Numericalsimulations--as opposed to exact symbolic calculations--indicate, on the otherhand, that the true probability is certainly less than 1/2. Our combinedanalyses lead us to suggest a possible form for the true DESF, yielding aseparability probability of 29/64 = 0.453125, while the best exact lower boundestablished so far is (6928-2205 Pi)/2^(9/2) = 0.0348338.

An improved exact upper bound of 22/35 on the Hilbert-Schmidt separability probability of real two-qubit systems