A Hypergeometric Formula Yielding Hilbert-Schmidt Generic 2 x 2 Generalized Separability Probabilities
Abstract
We significantly advance 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 here. It reproduces a series, alpha = 1/2, 1, 3/2, 2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability probabilities, advanced on the basis of systematic high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. For generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit, P(1) = 8/33, and (27-dimensional) two-quat(ernionic)bit systems, P(2)= 26/323. The function P(alpha) is generated--applying the Mathematica command FindSequenceFunction--using solely either) the thirty-two integral or half-integral probabilities, yet successfully predicts the unused thirty-two.
Keywords
Cite
@article{arxiv.1203.4498,
title = {A Hypergeometric Formula Yielding Hilbert-Schmidt Generic 2 x 2 Generalized Separability Probabilities},
author = {Paul B. Slater},
journal= {arXiv preprint arXiv:1203.4498},
year = {2012}
}
Comments
11 pages, 3 figures (Fig. 3 is the titular formula); slightly modified presentation; confirmation by C. Dunkl of quaternionic (alpha =2) moment formulas for n =1, 2 also noted (p. 2)