scholarly journals Entanglement witnesses arising from Choi type positive linear maps

2012 ◽  
Vol 45 (41) ◽  
pp. 415305 ◽  
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

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.

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.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

scholarly journals Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 313-324
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Positive Operators ◽  
Selfadjoint Operators ◽  
Popoviciu’S Inequality ◽  
Linear Maps ◽  
Superquadratic Functions ◽  
Operator Version ◽  
Positive Linear Maps

AbstractIn this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

scholarly journals All 2-positive linear maps from M3(C) to M3(C) are decomposable

2016 ◽  
Vol 503 ◽  
pp. 233-247 ◽  
Author(s):  
Yu Yang ◽  
Denny H. Leung ◽  
Wai-Shing Tang
Keyword(s):  
Linear Maps ◽  
Positive Linear Maps
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