scholarly journals Exposedness of Choi-Type Entanglement Witnesses and Applications to Lengths of Separable States

2013 ◽  
Vol 20 (04) ◽  
pp. 1350012 ◽  
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Large Class ◽  
Separable State ◽  
Matrix Algebras ◽  
Separable States ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

We present a large class of indecomposable exposed positive linear maps between 3 × 3 matrix algebras. We also construct two-qutrit separable states with lengths ten in the interior of their dual faces. With these examples, we show that the length of a separable state may decrease strictly when we mix it with another separable state.

scholarly journals FACIAL STRUCTURES FOR VARIOUS NOTIONS OF POSITIVITY AND APPLICATIONS TO THE THEORY OF ENTANGLEMENT

2013 ◽  
Vol 25 (02) ◽  
pp. 1330002 ◽  
Author(s):  
SEUNG-HYEOK KYE
Keyword(s):  
Convex Cones ◽  
Edge States ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, and 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.

scholarly journals Entanglement Witnesses Arising from Exposed Positive Linear Maps

2011 ◽  
Vol 18 (04) ◽  
pp. 323-337 ◽  
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Matrix Algebra ◽  
Three Dimensional ◽  
Entangled States ◽  
Entangled State ◽  
Minimal Set ◽  
Large Classes ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps ◽  
Ppt States

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.

Positive linear maps of matrix algebras

2012 ◽  
Vol 98 ◽  
pp. 293-302
Author(s):  
W. A. Majewski
Keyword(s):  
Matrix Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

Duality for positive linear maps in matrix algebras

2000 ◽  
Vol 86 (1) ◽  
pp. 130 ◽  
Author(s):  
Myoung-Hoe Eom ◽  
Seung-Hyeok Kye
Keyword(s):  
Matrix Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

Facial Structures for the Positive Linear Maps Between Matrix Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 74-82 ◽  
Author(s):  
Seung-Hyeok Kye
Keyword(s):  
Complete Lattice ◽  
Matrix Algebra ◽  
Convex Set ◽  
Matrix Algebras ◽  
Linear Maps ◽  
The Matrix ◽  
Positive Linear Maps

AbstractLet denote the convex set of all positive linear maps from the matrix algebra Mn(ℂ) into itself. We construct a join homomorphism from the complete lattice of all faces of into the complete lattice of all join homomorphisms between the lattice of all subspaces of ℂn . We also characterize all maximal faces of .

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