On the structure of entanglement witnesses and new class of positive indecomposable maps

We construct a new class of positive indecomposable maps in the algebra of 'd x d' complex matrices. Each map is uniquely characterized by a cyclic bistochastic matrix. This class generalizes a Choi map for d=3. It provides a new reach family of indecomposable entanglement witnesses which define important tool for investigating quantum entanglement.Comment: 18 page

Constructing new optimal entanglement witnesses

We provide a new class of indecomposable entanglement witnesses. In 4 x 4 case it reproduces the well know Breuer-Hall witness. We prove that these new witnesses are optimal and atomic, i.e. they are able to detect the "weakest" quantum entanglement encoded into states with positive partial transposition (PPT). Equivalently, we provide a new construction of indecomposable atomic maps in the algebra of 2k x 2k complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et. al.Comment: 9 page

Optimal entanglement witnesses from generalized reduction and Robertson maps

We provide a generalization of the reduction and Robertson positive maps in matrix algebras. They give rise to a new class of optimal entanglement witnesses. Their structural physical approximation is analyzed. As a byproduct we provide a new examples of PPT (Positive Partial Transpose) entangled states.Comment: 14 page

Geometry of entanglement witnesses for two qutrits

We characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of 3 x 3 complex matrices. It turns out that boundary of this set displays elegant representation in terms of SO(2) rotations. We conjecture that maps parameterized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parameterized by improper rotations from the orthogonal group O(2).Comment: 9 page

On structural physical approximations and entanglement breaking maps

Very recently a conjecture saying that the so-called structural physical approximations (SPAa) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed [J. K. Korbicz {\it et al.}, Phys. Rev. A {\bf 78}, 062105 (2008)]. The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses (EWs) supporting the conjecture. Then, we ask if SPAs constructed from other than the depolarizing channel maps also lead to EB maps and show that in general this is not the case. On the other hand, we prove an interesting fact that for any positive map $\Lambda$ there exists an EB channel $\Phi$ such that the SPA of $\Lambda$ constructed with the aid of $\Phi$ is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of construction of SPA and prove that in the case of the transposition map it gives EB channel.Comment: 22 pages, improved version, accepted by Journal of Physics

New tools for investigating positive maps in matrix algebras

We provide a novel tool which may be used to construct new examples of positive maps in matrix algebras (or, equivalently, entanglement witnesses). It turns out that this can be used to prove positivity of several well known maps (such as reduction map, generalized reduction, Robertson map, and many others). Furthermore, we use it to construct a new family of linear maps and prove that they are positive, indecomposable and (nd)optimal.Comment: 10 page