Extremals and exposed faces of the cone of positive maps

Yopp, David A.; Hill, R. D.

doi:10.1080/03081080412331272500

Cited by 17 publications

(17 citation statements)

References 19 publications

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“…It was shown in [20] (see also [33]) that φ 1 and φ 2 are exposed elements in the cones P(K 1 , H 1 ) and P(K 2 , H 2 ) respectively. Our aim is to show the following Theorem 4.1.…”

Section: Special Classes Of Mergingmentioning

confidence: 97%

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

Marciniak

Rutkowski

2017

Linear Algebra and its Applications

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For two positive maps φ i : B(K i ) → B(H i ), i = 1, 2, we construct a new linear map φ : B(H) → B(K), where K = K 1 ⊕ K 2 ⊕ C, H = H 1 ⊕ H 2 ⊕ C, by means of some additional ingredients such as operators and functionals. We call it a merging of maps φ 1 and φ 2 . The properties of this construction are discussed. In particular, conditions for positivity of φ, as well as for 2-positivity, complete positivity, optimality and nondecomposability, are provided. In particular, we show that for a pair composed of 2-positive and 2-copositive maps, there is a nondecomposable merging of them. One of our main results asserts, that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps, their canonical merging is an exposed positive map. This result provides a wide class of new examples of exposed positive maps. As an application, new examples of entangled PPT states are described.2010 Mathematics Subject Classification. Primary: 46L60, 15B48, 81P40; Secondary: 81Q10, 46L05.

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“…It was shown in [20] (see also [33]) that φ 1 and φ 2 are exposed elements in the cones P(K 1 , H 1 ) and P(K 2 , H 2 ) respectively. Our aim is to show the following Theorem 4.1.…”

Section: Special Classes Of Mergingmentioning

confidence: 97%

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

Marciniak

Rutkowski

2017

Linear Algebra and its Applications

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“…It is known that φ V also generates an exposed ray of the much bigger cone P 1 . See [131] and [90].…”

Section: Faces For Completely Positive Mapsmentioning

confidence: 99%

“…So, these are only candidates of exposed decomposable maps in the cone P 1 . It is known [131] that they are always extreme in P 1 , and exposed in the cone P 1 if the rank of V is one or full. More recently, it was shown in [90] that φ V is always exposed.…”

Section: Boundary Structures For Positive Mapsmentioning

confidence: 99%

Facial Structures for Various Notions of Positivity and Applications to the Theory of Entanglement

Kye

2013

Rev. Math. Phys.

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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.1991 Mathematics Subject Classification. 81P15, 15A30, 46L05.

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“…See [23], Theorem 18.6. In case of decomposable maps, the maps in (1) are known to be exposed [20,28], and there are no other exposed decomposable maps.…”

Section: Introductionmentioning

confidence: 99%

Indecomposable Exposed Positive Bi-linear Maps Between Two by Two Matrices

Kye

2018

Acta Math Vietnam

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Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between 2 × 2 matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose whose volume is nonzero.1991 Mathematics Subject Classification. 46L07, 15A30, 81P15.

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Extremals and exposed faces of the cone of positive maps

Abstract: We show that for a matrix A of size n Â m, the maps X°AXA Ã and X°AX T A Ã are extremals of the cone of positive maps acting C mÂm ! C nÂn . When the rank of A equals 1 or m (in which case, necessarily m nÞ, we show that these extremals are exposed.

Cited by 17 publications

References 19 publications

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

Facial Structures for Various Notions of Positivity and Applications to the Theory of Entanglement

Indecomposable Exposed Positive Bi-linear Maps Between Two by Two Matrices

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