Kil-Chan Ha

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.

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.

A class of atomic positive linear maps in matrix algebras

We find a large class of atomic positive linear maps between n-dimensional matrix algebras, and study their properties in connection with complete positivity, 2-positivity and Schwarz inequality.

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.

Geometry for separable states and construction of entangled states with positive partial transposes

We construct faces of the convex set of all

The structural physical approximations and optimal entanglement witnesses

2\otimes 4

Positive projections onto spin factors

bipartite separable states, which are affinely isomorphic to the simplex