Adam Rutkowski

Using non-positive maps to characterize entanglement witnesses

In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach crucial role play such maps which are surjective between sets

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

\mathcal{P}_{k}^d

Generalizing Choi-Like Maps

of

Construction and properties of a class of private states in arbitrary dimensions

k \leq d

Separable Decomposition and Quantum Correlations in Toeplitz Matrices

rank projectors and the set

Necessary and sufficient condition of separability for D-symmetric diagonal states.

\mathcal{P}_1^d

Paradoxical consequences of multipath coherence : perfect interaction-free measurements

of rank one projectors acting in the

Unified approach to geometric and positive-map-based nonlinear entanglement identifiers

d

Interaction-free measurements cannot be perfect

dimensional space. We argue that our method can be used to check whether a given observable is an entanglement witness. In the second part of this paper we show that inverse reduction map satisfies this requirement and using it we can obtain a bunch of new entanglement witnesses.

Saturation of the Tsirelson bound for the Clauser-Horne-Shimony-Holt inequality with random and free observables

Abstract For two positive maps ϕ i : B ( K i ) → B ( H i ) , i = 1 , 2 , we construct a new linear map ϕ : B ( H ) → B ( K ) , where K = K 1 ⊕ K 2 ⊕ C , H = H 1 ⊕ H 2 ⊕ C , by means of some additional ingredients such as operators and functionals. We call it a merging of maps ϕ 1 and ϕ 2 . The properties of this construction are discussed. In particular, conditions for positivity of ϕ, as well as for 2-positivity, complete positivity, optimality and indecomposability, are provided. In particular, we show that for a pair composed of 2-positive and 2-copositive maps, there is an indecomposable merging of them. One of our main results asserts, that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps, their canonical merging is an exposed positive map. This result provides a wide class of new examples of exposed positive maps. As an application, new examples of entangled PPT states are described.