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Formulas for Generalized Two-Qubit Separability Probabilities | | Paul B. Slater
; | | Date: |
27 Sep 2016 | | Abstract: | To begin, we find certain formulas $Q(k,alpha)= G_1^k(alpha)
G_2^k(alpha)$, for $k = -1, 0, 1,...,9$. These yield that part of the total
separability probability, $P(k,alpha)$, for generalized (real, complex,
quaternionic,ldots) 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, obtained by tracing over the pure states
in $4 imes (4 +k)$-dimensions, and $
ho^{PT}$, its partial transpose.
Further, $alpha$ is a Dyson-index-like parameter with $alpha = 1$ for the
standard (15-dimensional) convex set of (complex) two-qubit states. For $k=0$,
we obtain the previously reported Hilbert-Schmidt formulas, with (the real
case) $Q(0,frac{1}{2}) = frac{29}{128}$, (the standard complex case)
$Q(0,1)=frac{4}{33}$, and (the quaternionic case) $Q(0,2)=
frac{13}{323}$---the three simply equalling $ P(0,alpha)/2$. The factors
$G_2^k(alpha)$ are sums of polynomial-weighted generalized hypergeometric
functions $_{p}F_{p-1}$, $p geq 7$, all with argument $z=frac{27}{64}
=(frac{3}{4})^3$. We find number-theoretic-based formulas for the upper
($u_{ik}$) and lower ($b_{ik}$) parameter sets of these functions and, then,
equivalently express $G_2^k(alpha)$ in terms of first-order difference
equations. Applications of Zeilberger’s algorithm yield "concise" forms,
parallel to the one obtained previously for $P(0,alpha) =2 Q(0,alpha)$. For
nonnegative half-integer and integer values of $alpha$, $Q(k,alpha)$ has
descending roots starting at $k=-alpha-1$. Then, we (C. Dunkl and I) construct
a remarkably compact (hypergeometric) form for $Q(k,alpha)$ itself. The
possibility of an analogous "master" formula for $P(k,alpha)$ is, then,
investigated, and a number of interesting results found. | | Source: | arXiv, 1609.8561 | | Services: | Forum | Review | PDF | Favorites | |
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