Related papers: A class of exposed indecomposable positive maps

A map $\mathcal{P}$ is tensor stable positive (tsp) if $\mathcal{P}^{\otimes n}$ is positive for all $n$, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there…

The problem of classification of decomposable (in the sense of Stormer) positive maps between matrix algebras is presented. We propose the new notion of "finite" version of decomposability ($k$-decomposabilty). The characterisation of…

A new method of analysing positive bistochastic maps on the algebra of complex matrices $M_{3}$ has been proposed. By identifying the set of such maps with a convex set of linear operators on $\mathbb{R}^{8}$, one can employ techniques from…

In this thesis work, we have studied the role of positive and completely positive maps in detecting entanglement.

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…

We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on $M_2(C)$ and $M_3(C)$. Our main tool stems from classical Grothendieck…

We construct a family of map which is shown to be positive when imposing certain condition on the parameters. Then we show that the constructed map can never be completely positive. After tuning the parameters, we found that the map still…

In this paper, a class of indecomposable positive finite rank elementary operators of order $(n,n)$ are constructed. This allows us to give a simple necessary and sufficient criterion for separability of pure states in bipartite systems of…

We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic…

We introduce special classes of irreducible representations of groups: thick representations and dense representations. Denseness implies thickness, and thickness implies irreducibility. We show that absolute thickness and absolute…

We provide a straightforward generalization of a positive map in $M_3(\mathbb{C})$ considered recently by Miller and Olkiewicz \cite{Miller}. It is proved that these maps are optimal and indecomposable. As a byproduct we provide a class of…

This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the…

Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot…

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…

A map $\phi:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form $\phi=\phi_1+\phi_2$ where $\phi_1$ is a CP map while $\phi_2$ is a co-CP map. It is known that if $m=n=2$ then every positive map is decomposable. Given an extremal…

In this article we describe indecomposable objects of the derived categories of a branch class of associative algebras. To this class belong such known classes of algebras as gentle algebras, skew-gentle algebras and certain degenerations…

From the physical point of view entanglement witnesses define a universal tool for analysis and classification of quantum entangled states. From the mathematical point of view they provide highly nontrivial generalization of positive…

We outline a new approach to the characterization as well as to the classification of positive maps. This approach is based on the facial structures of the set of states and of the cone of positive maps. In particular, the equivalence…

We study mapping cones and their dual cones of positive maps of the n by n matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular…

We construct certain non-degenerate maps and sets, mainly in the complex-analytic category. For example, we show that for every countable subset S in an irreducible complex space X there exists a holomorphic map from the unit disk to X such…