Advances in delimiting the Hilbert–Schmidt separability probability of real two-qubit systems

Abstract

We seek to derive the probability—expressed in terms of the Hilbert–Schmidt (Euclidean or flat) metric—that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres–Horodecki test on the partial transposes (PTs) of the associated 4 × 4 density matrices (ρ). But the full implementation of the test—requiring that the determinant of the PT be nonnegative for separability to hold—appears to be, at least presently, computationally intractable. So, we have previously implemented—using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)—the weaker implied test of nonnegativity of the six 2 × 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of . Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 × 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of . Then, we conclude that a still further improved upper bound of can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 × 3 principal minors. Numerical simulations—as opposed to exact symbolic calculations—indicate, on the other hand, that the true probability is certainly less than . Our analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of , while the absolute separability probability of provides the best exact lower bound established so far. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESFs that comes close to reproducing our numerical estimate of the true separability function.