Thursday, July 4, 2013

1301.6617 (Paul B. Slater)

A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability

Paul B. Slater
We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha), incorporating a family of six hypergeometric functions, all with argument 27/64 = (3/4)^3, is obtained. It reproduces a series, alpha = 1/2, 1, 3/2,...,32 of sixty-four conjectured Hilbert-Schmidt rational-valued generic 2 x 2 separability probabilities. These exact ratios are put forth on the basis of systematic, high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. A lengthy expression for P(alpha) containing six generalized hypergeometric functions is initially obtained--making use of the FindSequenceFunction command of Mathematica. A remarkably succinct re-expression for P(alpha) is then found, by Qing-Hu Hou and colleagues, using Zeilberger's algorithm ("creative telescoping"), For generic (9-dimensional) two-rebit systems, P(1/2}) = 29/64, (15-dimensional) two-qubit, P(1) = 8/33 (a value that had been proposed in J. Phys. A, 40, 14279 [2007] and supported in both Intl. J. Mod. Phys. B, 26, 1250054 [2012] and Phys. Rev. A, 86, 042325 [2012]) and (27-dimensional) two-quater(nionic)bit systems, P(2)=26/323.
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