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 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.

On the convex set of completely positive linear maps in matrix algebras

1997 ◽  
Vol 122 (1) ◽  
pp. 45-54 ◽  
Author(s):  
SEUNG-HYEOK KYE
Keyword(s):  
Convex Compact ◽  
Compact Set ◽  
Completely Positive ◽  
One Dimensional ◽  
Matrix Algebras ◽  
Linear Maps ◽  
The Matrix ◽  
Rank One ◽  
Convex Compact Set ◽  
Positive Linear Maps

Let PI (respectively CPI) be the convex compact set of all unital positive (respectively completely positive) linear maps from the matrix algebra Mm([Copf ]) into Mn([Copf ]). We show that maximal faces of CPI correspond to one dimensional subspaces of the vector space Mm, n([Copf ]). Furthermore, a maximal face of CPI lies on the boundary of PI if and only if the corresponding subspace is generated by a rank one matrix.

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 Diagonal unitary and orthogonal symmetries in quantum theory

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 519
Author(s):  
Satvik Singh ◽  
Ion Nechita
Keyword(s):  
Sufficient Conditions ◽  
Quantum Channels ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Completely Positive Matrices ◽  
Linear Maps ◽  
Positive Matrices ◽  
Invariant States ◽  
Class Of Matrices ◽  
Necessary And Sufficient

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

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 Extremal problems for completely positive maps

1994 ◽  
Vol 17 (3) ◽  
pp. 607-608
Author(s):  
Mingze Yang
Keyword(s):  
Extremal Problems ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Convex Subsets

In this note, we study the faces of some convex subsets ofCPc(A,B(ℋ))(the continuous completely positive linear maps from pro-C*-algebraAtoB(ℋ)).

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