Rank properties of exposed positive maps

Marciniak, Marcin

doi:10.48550/arxiv.1103.3497

Cited by 7 publications

(15 citation statements)

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“…Remark. This corollary immediately follows from results in [20] and repeats the result already given in [3] (which was obtained via convex analysis). Also the criterion given in [21] shows that transposition map is an exposed map.…”

Section: Symmetries As Exposed Points Of Dsupporting

confidence: 82%

On the structure of positive maps. II. Low dimensional matrix algebras

Majewski

Tylec

2013

Journal of Mathematical Physics

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We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on M 2 ( ) and M 3 ( ). Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiołkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting we show that in M 2 ( ) case a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, 2P x with x maximally entangled vector and p⊗½ with p rank 1 projector. We show why this construction fails in M 3 ( ) case. There are also similarities. In both M 2 ( ) and M 3 ( ) cases any unital positive map represented by selfadjoint unitary is unitarily equivalent to the transposition map. Consequently we obtain a large family of exposed maps. We also investigate a convex structure of the Choi map, the first example of non-decomposable map. As a result the nature of the Choi map will be explained. This gives an information on the origin of appearance of non-decomposable maps on M 3 ( ).

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Section: Symmetries As Exposed Points Of Dsupporting

confidence: 82%

On the structure of positive maps. II. Low dimensional matrix algebras

Majewski

Tylec

2013

Journal of Mathematical Physics

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“…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%

“…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%

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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“…and so the set P φ V consist of all product vectors orthogonal to V . If nonzero V is not of rank one, then Lemma 2.3 of [30] tells us that the pair (V ⊥ , {0} ⊥ ) satisfies the range criterion. This is equivalent to say that both φ V and φ V generate rays in the cone of D which are exposed by separable states by [27].…”

Section: Ppt Entanglement Detected By Exposed Indecomposable Mapsmentioning

confidence: 99%

“…It was shown in [44] and [30] that the map φ V is exposed for each matrix V . It is now clear that the maps φ V and φ V exhaust all decomposable positive linear maps which generate exposed rays of P 1 .…”

Section: Introductionmentioning

confidence: 99%

Entanglement Witnesses Arising from Exposed Positive Linear Maps

Kye

2011

Open Syst. Inf. Dyn.

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We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entanglement can be detected by one of these witnesses, and this witness detects a unique set of entanglement among those. Therefore, they provide a minimal set of witnesses to detect all entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entanglement 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 between three dimensional matrix algebra.1991 Mathematics Subject Classification. 81P15, 15A30, 46L05. Therefore, we have D[α, p, q, r] ≤ (t − 1) 2 (t 2 + t + 1)(t 2 + 1)T 3 + (t − 1) 2 (t 2 + t + 1)t(−2T 3 ) = (t − 1) 4 (t 2 + t + 1)T 3 ≤ 0, 13

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Rank properties of exposed positive maps

Cited by 7 publications

References 0 publications

On the structure of positive maps. II. Low dimensional matrix algebras

On the structure of positive maps. II. Low dimensional matrix algebras

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

Entanglement Witnesses Arising from Exposed Positive Linear Maps

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