Seung-Hyeok Kye

Generalized Choi maps in three-dimensional matrix algebra☆

We consider a class of positive linear maps in the three-dimensional matrix algebra, which are generalizations of the positive linear map constructed by Choi in the relation with positive semidefinite biquadratic forms. We find conditions for which such maps are completely positive, completely copositive, decomposable, and two-positive.

Entangled states with positive partial transposes arising from indecomposable positive linear maps

We construct entangled states with positive partial transposes using indecomposable positive linear maps between matrix algebras. We also exhibit concrete examples of entangled states with positive partial transposes arising in this way, and show that they generate extreme rays in the cone of all positive semi-definite matrices with positive partial transposes. They also have Schmidt numbers two.

FACIAL STRUCTURES FOR VARIOUS NOTIONS OF POSITIVITY AND APPLICATIONS TO THE THEORY OF ENTANGLEMENT

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.

Entanglement Witnesses Arising from Exposed Positive Linear Maps

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.

Construction of entangled states with positive partial transposes based on indecomposable positive linear maps

We show that every entanglement with positive partial transpose may be constructed from an indecomposable positive linear map between matrix algebras.

Construction of 3 ⊗ 3 entangled edge states with positive partial transposes

We construct a class of 3 ⊗ 3 entangled edge states with positive partial transposes using indecomposable positive linear maps. This class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes.

Classification of bi-qutrit positive partial transpose entangled edge states by their ranks

We construct 3 ⊗ 3 PPT entangled edge states with maximal ranks, to complete the classification of 3 ⊗ 3 PPT entangled edge states by their types. The ranks of the states and their partial transposes are 8 and 6, respectively. These examples also disprove claims in the literature.

FACIAL STRUCTURES FOR SEPARABLE STATES

The convex cone V1 generated by separable states is contained in the cone T of all positive semi-definite block matrices whose block transposes are also positive semi-definite. We characterize faces of the cone V1 induced by faces of the cone T which are determined by pairs of subspaces of matrices.

Entanglement witnesses arising from Choi type positive linear maps

We construct optimal PPTES witnesses to detect 3⊗3 PPT entangled edge states of type (6, 8) constructed recently by Kye and Osaka (2012 J. Math. Phys. 53 052201). To do this, we consider positive linear maps which are variants of the Choi type map involving complex numbers, and examine several notions related to the optimality of those entanglement witnesses. Throughout the discussion, we suggest a method to check the optimality of entanglement witnesses without the spanning property.

Exposedness of Choi-Type Entanglement Witnesses and Applications to Lengths of Separable States

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.