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Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure | | Paul B. Slater
; | | Date: |
17 Apr 2015 | | Abstract: | We find equivalent hypergeometric- and difference-equation-based formulas,
$Q(k,alpha)= G_1^k(alpha) G_2^k(alpha)$, for $k = -1, 0, 1,ldots,9$, for
that (rational-valued) portion of the total separability probability for
generalized two-qubit states endowed with random induced measure, for which the
determinantal inequality $|
ho^{PT}| >|
ho|$ holds. Here $
ho$ denotes a $4
imes 4$ density matrix and $
ho^{PT}$, its partial transpose, while $alpha$
is a Dyson-index-like parameter with $alpha = 1$ for the standard
(15-dimensional) convex set of two-qubit states. The dimension of the space in
which these density matrices is embedded is $4 imes (4 +k)$. For the
symmetric case of $k=0$, we obtain the previously reported Hilbert-Schmidt
formulas, with (the two-re[al]bit case) $Q(0,frac{1}{2}) = frac{29}{128}$,
(the standard two-qubit case) $Q(0,1)=frac{4}{33}$, and (the
two-quater[nionic]bit case) $Q(0,2)= frac{13}{323}$. The factors
$G_2^k(alpha)$ can be written as the sum of weighted hypergeometric functions
$_{p}F_{p-1}$, $p geq 7$, all with argument $frac{27}{64} =(frac{3}{4})^3$.
We find formulas for the upper and lower parameter sets of these functions and,
then, equivalently express $G_2^k(alpha)$ in terms of first-order difference
equations. The factors $G_1^k(alpha)$ are equal to
$(frac{27}{64})^{alpha-1}$ times ratios of products of six Pochhammer symbols
involving the indicated parameters. Some remarkable $alpha-$ and $k$-specific
invariant asymptotic properties (again, involving $frac{27}{64}$ and related
quantities) of separability probability formulas emerge. | | Source: | arXiv, 1504.4555 | | Services: | Forum | Review | PDF | Favorites | |
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