The norms of Bloch vectors and classification of four qudits quantum states

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits quantum systems.Comment: 8 pages, 1 figur

Research and Practice of Quantitative Models on The Freedom of Trade Based on AHP-FCE Method: Taking Major Countries in WTO as Examples

In recent years, protectionism and unilateralism in the field of international trade are increasing. The WTO-centric multilateral trade regime has been hit. By researching the index of trade freedom among countries in international trade, this paper takes five important countries in WTO as an example, and constructs the trade freedom index and the national trade freedom model. The study reviewed and analysed global and regional developments in cross-border trade and investigated five varialbes that impacted the index of trade freedom. Then the fuzzy comprehensive analysis method (FCE) is used to obtain the comprehensive evaluation matrix by fuzzy arithmetic processing through the evaluation index weight and fuzzy relation matrix. Therefore, the changing data of the index of trade freedom in five countries from 2017 to 2018 are obtained. It is discovered that there is a significant difference in the index of trade freedom among countries. What’s worse, the average of the global index has declined year by year. This phenomenon is caused by the decline of global trade volume and the prevalence of new trade barriers. In the end, this paper proposes the implementation of multilateral negotiations and the establishment of long-term penalties as possible solutions