2,054 research outputs found
Necessary conditions for optimality of decomposable entanglement witnesses
It is well known that the support of an optimal decomposable entanglement
witness is completely entangled. We add two more necessary conditions for the
optimality: The orthogonal complement of the support must have a nonzero
product vector; another one will be given in terms of related faces of a convex
cone. With these necessary conditions, we show that there exist examples of
non-optimal decomposable entanglement witnesses which are the partial
transposes of positive semi-definite matrices supported on completely entangled
spaces, whenever both of the local dimensions are greater than or equal to
three.Comment: Final form to be published in Rep. Math. Phys. The last example is
corrected together with minor text
On exposed positive maps: Robertson and Breuer-Hall maps
It is well known that so called Breuer-Hall positive maps used in
entanglement theory are optimal. We show that these maps possess much more
subtle property --- they are exposed. As a byproduct it proves that a Robertson
map in the algebra of 4 x 4 complex matrices is not only extreme, which was
already shown by Robertson, but also exposed.Comment: 7 page
Entanglement witnesses arising from Choi type positive linear maps
We construct optimal PPTES witnesses to detect PPT entangled
edge states of type constructed recently \cite{kye_osaka}. To do this,
we consider positive linear maps which are variants of the Choi type map
involving complex numbers, and examine several notions related to optimality
for those entanglement witnesses. Through the discussion, we suggest a method
to check the optimality of entanglement witnesses without the spanning
property.Comment: 18 pages, 4 figures, 1 tabl
Facial structures for various notions of positivity and applications to the theory of entanglement
In this expository note, we explain facial structures for the convex cones
consisting of positive linear maps, completely positive linear maps,
decomposable positive linear maps between matrix algebras, respectively. These
will be applied to study the notions of entangled edge states with positive
partial transposes and optimality of entanglement witnesses.Comment: An expository note. Section 7 and Section 8 have been enlarge
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