Abstract
In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map 
can be written as the difference between two completely positive maps Λ=Λ1−Λ2, we propose a possible way to generalize the Nielsen–Kempe majorization criterion. Then, we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of Schur-concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen–Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map Λ, then there exist entropic inequalities derived from Λ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [I⊗Λ](ϱAB)⩾0. Moreover, our inequalities provide a way of constructing multi-copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of the recently introduced protocol of state merging and the possibility of approximating the mean value of a linear entanglement witness.
Export citation and abstract BibTeX RIS
GENERAL SCIENTIFIC SUMMARY Introduction and background. Entanglement is one of the most interesting topics in modern physics. It has become a basic resource for (quantum) information processing with quantum systems and made feasible such tasks as establishing a secure cryptographic key between distant laboratories or teleportation of particles. However to determine if two quantum systems are entangled and consequently useful for processing of quantum information is a hard theoretical, and even harder experimental problem. Therefore in our research we concentrated on development of criteria that allow for detection of quantum correlations and at the same time are experimentally friendly.
Main results. In our approach we focused on development of quantum observables that can be measured on several copies of a quantum state at a time in a so called collective measurement. If the average outcome of the measurement of such an observable is negative then we are certainly dealing with an entangled state. In our paper we give a general method allowing for construction of these types of observables and analyse their effectiveness. A big advantage of this approach is the universality of the criteria, meaning that the potential experimental entanglement test can be performed efficiently without any previous knowledge about the analysed state.
Wider implications. The qualitative methods of entanglement detection which were presented in our paper are designed to answer only the question of whether a given state is entangled. The next step is to see if a similar approach would be useful in quantifying entanglement, that is determining the extent to which a given state is entangled.