Dualities and positivity in the study of quantum entanglement

We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank. In particular, we prove that separable states of length lower of equal 3 have Schmidt ranks equal to their lengths. We also give an example of a state which has length 4 and Schmidt rank 3.Comment: A shortened and amended version of author's Master's Thesi

Three-point Nevanlinna Pick problem in the polydisc

It is very elementary to observe that functions interpolating an extremal two-point Pick problem on the polydisc are just left inverses to complex geodesics. In the present article we show that the same property holds for a three-point Pick problem on polydiscs, i.e. it may be expressed it in terms of three-complex geodesics. Using this idea we are able to solve that problem obtaining formulas and a uniqueness theorem for solutions of extremal problems. In particular, we determine a class of rational inner functions interpolating that problem. Possible extensions and further investigations are also discussed

Development planning versus participation of inhabitants in management

Purpose: There is an emphasis that planning sustainable development is conditioned by external and internal factors. The basis for such planning is a range of an area research which can distinguish the groups of similar units and quantify their stages of development. The aim of the article is to present the author’s methodological proposal in the field of management and development planning, taking the opinions of the commune inhabitants. Design/Methodology/Approach: The paper presents the application of changes in planning the sustainable development, in the context of a stable economy. In present conditions, there is a lack of problem identification, especially social and economic. The proposed procedure of methodology on the basis of inhabitants’ opinions of the communes of Swietokrzyskie Voivodeship situated in protected areas, such as landscape parks and regions of protected landscape is an element of a wider research problem. At present this problem has and is going to have a more significant meaning, the participation of local communities in creating their own development. Findings: The problem has already had and will continue to have a growing importance, because it will no longer be possible to prevent the participation of local societies in creating their own development. Practical Implications: The realisation of sustainable development will require a total change of the orientation of all the parties interested, according to the social and economic problems (basic principles). Originality/Value: In the majority of the recommended planning procedures, there is a possibility of economy improvement and contribution to the improvement of social and economic welfare.peer-reviewe