A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15-and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are . We, now, set the in the complexcase-conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory-we take S complex (ν) proportional to S 2 real (ν). We also investigate the real and complex qubit-qutrit cases. Now, there are two variables, ν 1 =

Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |ρ| k |ρ P T | n , k, n = 0, 1, 2, 3, . . ., P T denoting partial transpose, for both generic (9-dimensional) two-rebit (α = 1 2 ) and generic (15-dimensional) two-qubit (α = 1) density matrices ρ. The results are, then, incorporated by Dunkl into a general formula (Appendix D 6), parameterized by k, n and α, with the case α = 2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter α fixed, the induced univariate moments (|ρ||ρ P T |) n and |ρ P T | n are inputted into a Legendre-polynomialbased (least-squares) probability-distribution reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding α-specific separability probability estimates. Since, as the number of inputted moments grows, estimates based on the variable |ρ||ρ P T | strongly decrease, while ones employing |ρ P T | strongly increase (and converge faster), the gaps between upper and lower estimates diminish, yielding sharper and sharper bounds. Remarkably, for α = 2, with the use of 2,325 moments, a separability-probability lower-bound 0.999999987 as large as 43, 195302 (2010)).

Due to recent important work of Życzkowski and Sommers ͓J. Phys. A 36, 10115 ͑2003͒; 36, 10083 ͑2003͔͒, exact formulas are available ͑in terms of both the Hilbert-Schmidt and the Bures metrics͒ for the ͑n 2 −1͒-dimensional and ͓n͑n −1͒ /2−1͔-dimensional volumes of the complex and real n ϫ n density matrices. However, no comparable formulas are available for the volumes ͑and, hence, probabilities͒ of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n = 4, that is, two-qubit systems. Making use of the density matrix ͑͒ parametrization of Bloore ͓J. Phys. A 9, 2059 ͑1976͔͒, we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, = 11 44 / 22 33 . The associated integrand in each case is the product of a known Jacobian ͑highly oscillatory near =1͒ and a certain unknown univariate function, which our extensive numerical ͑quasi Monte Carlo͒ computations indicate is very closely proportional to an ͑incomplete͒ Beta function B ͑a , b͒, with a =1/2, b = ͱ 3 in the real case, and a =2 ͱ 6/5,b =3/ ͱ 2 in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.

We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 × 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability probability function P (α) is put forth, yielding for generic (9-dimensional) two-rebit systems, P ( 1 2 ) = 29 64 , (15-dimensional) two-qubit systems, P (1) = 8 33 and (27-dimensional) two-quater(nionic)bit systems, P (2) = 26 323 . This particular form of P (α) was obtained by Qing-Hu Hou by applying Zeilberger's algorithm ("creative telescoping") to a fully equivalent-but considerably more complicated-expression containing six 7 F 6 hypergeometric functions (all with argument 27 64 = ( 3 4 ) 3 ). That hypergeometric form itself had been obtained using systematic, high-accuracy probability-distribution-reconstruction computations. These employed 7,501 determinantal moments of partially transposed 4 × 4 density matrices, parameterized by α = 1 2 , 1, 3 2 , 2, . . . , 32. From these computations, exact rational-valued separability probabilities were discernible. The (integral/half-integral) sequences of 32 rational values, then, served as input to the Mathematica FindSequenceFunction command, from which the initially obtained hypergeometric form of P (α) emerged.

Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a "natural measure" on the N -dimensional quantum systems, but expressed surprise when it led them to conclude that for N = 2 × 2, disentangled (separable) systems are more probable (0.632 ± 0.002) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 × 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis -which yields the smallest probabilities of disentanglement -uses the minimum number of parameters possible, that is N 2 − 1, as opposed to N 2 + N − 1 (although this "over-parameterization", as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities and the need for additional (possibly supercomputer) analyses is indicated -particularly so for higher-dimensional quantum systems (such as the 2 × 3 ones, we also study here). 89.70.+c, 02.40.Ky PACS Numbers