arXiv:1111.2012 · Journal of Physics A Mathematical and Theoretical

Exposed positive maps: a sufficient condition

Dariusz Chruściński, Gniewomir Sarbicki

Published Mar 1, 2012• 24 citations

Abstract

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a sufficient condition for a positive map to be exposed. This is an analogue of a spanning property which guarantees that a positive map is optimal. We analyze a class of decomposable maps for which this condition is also necessary.

DOI: 10.1088/1751-8113/45/11/115304

Related papers

A class of exposed indecomposable positive maps

Gniewomir Sarbicki, Dariusz Chruściński

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a class of indecomposable positive maps in the algebra of 2 n × 2 n complex matrices with n ⩾ 2. It is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states.

arXiv:1201.5995Journal of Physics A Mathematical and TheoreticalDec 5, 201219 citations

On exposed positive maps: Robertson and Breuer-Hall maps

Dariusz Chruściński

It is well known that so called Breuer-Hall positive maps used in entanglement theory are optimal. We show that these maps possess much more subtle property --- they are exposed. As a byproduct it proves that a Robertson map in the algebra of 4 x 4 complex matrices is not only extreme, which was already shown by Robertson, but also exposed.

arXiv:1108.2233Aug 10, 20112 citations

A decomposition theorem for positive maps, and the projection onto a spin factor

Erling Størmer

It is shown that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal. The result is analyzed in detail for the positive projection onto a spin factor.

arXiv:1308.3332Aug 15, 20131 citation

Decomposability and Optimality of a Class of Positive Maps

Jinchuan Hou

NCP positive linear maps play important rules in detecting entanglement of states in composite quantum systems.A kind of NCP positive maps are studied.It is shown that these positive linear maps Φ are decomposable and not 2-positive.A necessary and sufficient condition is given for the entanglement witnesses W Φ arising from such Φ to be optimal.

Journal of Shanxi UniversityJan 1, 20120 citations

Exposedness of elementary positive maps between matrix algebras

Seung-Hyeok Kye

The positive linear maps Ads which send matrices x to s∗xs play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ads generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

arXiv:2301.05788Linear and Multilinear AlgebraJan 18, 20242 citations

Exposed Positive Maps in M4(ℂ)

Dariusz Chruściński, Gniewomir Sarbicki

It is shown that the family of so-called Breuer-Hall maps in M4(ℂ) possesses strong spanning property and hence they are exposed in the convex cone of positive maps in M4(ℂ).

arXiv:1204.3109Open Systems & Information DynamicsSep 1, 201218 citations