Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments

Abstract: We seek to gain insight into the nature of the determinantal moments of generic (9-dimensional) two-rebit and (15-dimensional) two-qubit systems (rho). Such information-as it has proved to be in the Hilbert-Schmidt counterpart--should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining improved, or possibly even exact Bures 2 x 2 separability probabilities for such systems. The (regularizing) strategy we first adopt is to plot the ratio of numerically-generated (Ginibre ensemble) estimates of the Bures moments to the corresponding (apparently) exactly-known Hilbert-Schmidt moments (J. Phys. A, 45, 095305 [2012]). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures moments. In particular, the first moment (average) of |rho^{PT}| (where |rho^{PT}| in [-1/16, 1/256]), PT denoting partial transpose, for the two-qubit systems is, remarkably, -1 / 256. The analogous value for the two-rebit systems is -2663/(2^13 x 3 x 5 x 7) approx -0.00309594. While, an important function in the Hilbert-Schmidt case is the ratio of 3n-degree polynomials, it appears to be the ratio of 5n-degree polynomials in the Bures case.