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Bound-Entangled Probabilities for Magic Simplices | | Paul B. Slater
; | | Date: |
22 May 2019 | | Abstract: | We examine the $d=3$ (two-qutrit) and $d=4$ (two-qudit) subsets analyzed by
Hiesmayr and Loffler of the "magic" simplices of Bell states that were
introduced by Baumgartner, Hiesmayr and Narnhofer. The Hilbert-Schmidt
probabilities of positive-partial-transpose (PPT) states--within which
bound-entangled states, if any, must lie--are $frac{8 pi }{27 sqrt{3}}
approx 0.537422$ ($d=3$) and $frac{1}{24} left(12+sqrt{3} log
left(2-sqrt{3}
ight)
ight) approx 0.404957$ ($d=4$). Employing certain
entanglement witnesses, we obtain bound-entangled probabilities of
$-frac{4}{9}+frac{4 pi }{27 sqrt{3}}+frac{log (3)}{6} approx 0.00736862$
and $frac{-204+7 log (7)+168 sqrt{3} cos
^{-1}left(frac{11}{14}
ight)}{1134} approx 0.0032561229$ ($d=3$)--a general
formula for which is constructed by N. Tessore--and $frac{8 log
(2)}{27}-frac{59}{288} approx 0.00051583$ and $frac{24
ext{csch}^{-1}left(frac{8}{sqrt{17}}
ight)}{17 sqrt{17}}-frac{91}{544}
approx 0.00218722$ ($d=4$). In the $d=3$, case, we first utilized the mutually
unbiased bases (MUB) test that Hiesmayr and Loffler applied, and also the Choi
entanglement witness $W^{(+)}$ test. The same bound-entangled probability was
achieved with both. Their union doubles the probability, since the intersection
of the two sets of bound-entangled states detected is empty. If the
PPT-requirement is not imposed, the entanglement (bound and "non-bound")
probability for each test is $frac{1}{6} approx 0.16667$, while their union
and intersection yield probabilities of $frac{2}{9} approx 0.222222$ and
$frac{1}{9} approx 0.11111$. Also, we estimate PPT-probabilities of
approximately 0.39339 and 0.1157 for the original (8- [two-qutrit] and 15
[two-qudit]-dimensional) magic simplices themselves. | | Source: | arXiv, 1905.9228 | | Services: | Forum | Review | PDF | Favorites | |
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