1109.2560 (Paul B. Slater)

Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic
2 x 2 Separability Probabilities    [PDF]

Paul B. Slater

Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number
of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n,
k,n=0,1,2,3,..., PT denoting partial transpose, for both generic
(9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit
(alpha=1) density matrices rho. The results are, then, incorporated by Dunkl
into a general formula (Appendix D6), parameterized by k, n and alpha, with the
case alpha=2, presumptively corresponding to generic (27-dimensional)
quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the
induced univariate moments (|rho| |rho^{PT}|)^n and |rho^{PT}|^n are inputted
into a Legendre-polynomial-based (least-squares) probability-distribution
reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding
alpha-specific separability probability estimates. Since, as the number of
inputted moments grows, estimates based on |rho| |rho^{PT}| strongly decrease,
while ones employing |rho^{PT}| strongly increase (and converge faster), the
gaps between upper and lower estimates diminish, yielding sharper and sharper
bounds. Remarkably, for alpha = 2, with the use of 2,325 moments, a
separability-probability lower-bound 0.999999987 as large as 26/323 = 0.0804954
is found. For alpha=1, based on 2,415 moments, a lower bound results that is
0.999997066 times as large as 8/33 = 0.242424, a (simpler still) fractional
value that had previously been conjectured (J. Phys. A, 40, 14279 (2007)).
Furthermore, for alpha = 1/2, employing 3,310 moments, the lower bound is
0.999955 times as large as 29/64 = 0.453125, a rational value previously
considered (J. Phys. A, 43, 195302 (2010)).

View original: http://arxiv.org/abs/1109.2560