Wednesday, March 21, 2012

1203.4498 (Paul B. Slater)

A Hypergeometric Formula for Hilbert-Schmidt Generic 2 x 2 Generalized
Separability Probabilities

Paul B. Slater
We pursue the research agenda set forth in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). But in a more thorough, systematic manner, employing--for probability-distribution reconstruction purposes--a substantially greater number (7,501) of moments of the determinant |rho^{PT}| of the partial transpose rho^{PT} of the corresponding 4 x 4 density matrix rho. The results strengthen the conjectures that the two-rebit (alpha= 1/2) and two-qubit (alpha = 1) separability probabilities are 29/64 and 8/33, respectively. Additionally fortified is the conjecture that the presumptive quaternionic (alpha= 2) analog is 26/323. With high accuracy calculations conducted jointly for the sixty-four values alpha = 1/2, 1, 3/2, 2,...,32, we are able to obtain a certain (hypergeometric-related) function of alpha (with argument z = (3/4)^3 = 27/64) that successfully reproduces the three specific conjectures. Additionally, it yields for the other half-integral and integral values of alpha, rational-valued "generalized separability probabilities", matching to high precision the numerical estimates.
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