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A Hypergeometric Formula for Hilbert-Schmidt Generic 2 x 2 Generalized Separability Probabilities | Paul B. Slater
; | Date: |
20 Mar 2012 | Abstract: | 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. | Source: | arXiv, 1203.4498 | Services: | Forum | Review | PDF | Favorites |
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