We show that all of the known generalizations Choi maps are atomic maps.

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. show that every can be detected by one of these witnesses, and this witness detects a unique set among those. Therefore, they provide minimal to detect all in sense. Furthermore, if those are indecomposable then large classes with partial transposes have nonempty relative interiors the cone generated PPT states. also parameter family This gives first examples such between three...

We examine various notions related with the optimality for entanglement witnesses arising from Choi type positive linear maps. found examples of optimal which are non-decomposable, but not `non-decomposable witnesses' in sense [M. Lewenstein, B. Kraus, J. Cirac, and P. Horodecki, Phys. Rev. A 62, 052310 (2000)]. suggest to use term `PPTES witness' `optimal PPTES places order avoid possible confusion. also non-extremal indecomposable.

In the recent paper [Chru\'{s}ci\'{n}ski and Wudarski, arXiv:1105.4821], it was conjectured that entanglement witnesses arising from some generalized Choi maps are optimal. We show this conjecture is true. Furthermore, we they provide a one parameter family of indecomposable optimal witnesses.

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

We construct faces of the convex set all $2\ensuremath{\bigotimes}4$ bipartite separable states, which are affinely isomorphic to simplex ${\ensuremath{\Delta}}_{9}$ with 10 extreme points. Every interior point these is a state has unique decomposition into product even though ranks and its partial transpose 5 7, respectively. also note that number greater than $2\ifmmode\times\else\texttimes\fi{}4$, disprove conjecture on lengths qubit-qudit states. This face inscribed in corresponding PPT...

We present a large class of indecomposable exposed positive linear maps between three dimensional matrix algebras. also construct two qutrit separable states with lengths ten in the interior their dual faces. With these examples, we show that length state may decrease strictly when mix it another state.

We construct optimal PPTES witnesses to detect $3\otimes 3$ PPT entangled edge states of type $(6,8)$ constructed recently \cite{kye_osaka}. To do this, we consider positive linear maps which are variants the Choi map involving complex numbers, and examine several notions related optimality for those entanglement witnesses. Through discussion, suggest a method check without spanning property.

We introduce the notions of positive and copositive types for entanglement witnesses, depending on distance to part part. An witness $W$ is type if only its partial transpose $W^\Gamma$ type. show that structural physical approximation separable then should be type, SPA never unless both This shows conjecture meaningful those provide examples fails even case types.

We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding have unique decomposition, and show that this is true in most cases if the number sufficiently small. In three qubit case, generic five dimensional spaces give rise to faces convex consisting all states, are affinely isomorphic simplex six vertices. As byproduct, we construct entangled PPT edge rank four explicit formulae. This covers those entanglement cannot be...

Chru\ifmmode \acute{s}\else \'{s}\fi{}ci\ifmmode \acute{n}\else \'{n}\fi{}ski, Jurkowski, and Kossakowski [Phys. Rev. A 77, 022113 (2008)] studied quantum states with strong positive partial transpose (SPPT) conjectured that all SPPT are separable. We construct a two-parameter class of $3\ensuremath{\bigotimes}3$ entangled states, so the conjecture does not hold true for general states.

We construct a large class of indecomposable positive linear maps from the matrix algebra into algebra, which generate exposed extreme rays convex cone all maps. show that points dual faces for separable states arising these are parametrized by Riemann sphere, and hulls circle parallel to equator have exactly same properties with hull trigonometric moment curve studied combinatorial topology. Any interior boundary full ranks. exhibit concrete examples such states.

We construct triqubit genuinely entangled states which have positive partial transposes (PPTs) with respect to the bipartition of systems. These examples disprove a conjecture [Novo, Moroder, and G\"uhne, Phys. Rev A 88, 012305 (2013)] claims that PPT mixtures are necessary sufficient for biseparability three qubits.

One of the interesting problems on optimal indecomposable entanglement witnesses is whether there exists an witness which neither has spanning property nor associated with extremal positive linear map. Here, we answer this question negatively by examining extremality maps constructed Qi and Hou [J. Phys. A {\bf 44}, 215305 (2100)].

We show that all $2\ensuremath{\bigotimes}4$ states with strong positive partial transposes (SPPTs) are separable. also construct a family of $2\ensuremath{\bigotimes}5$ entangled SPPT states, so the conjecture on separability is completely settled. In addition, we clarify relation between set $2\ensuremath{\bigotimes}d$ separable and for case $d=3,4$.