Necessary and sufficient conditions for bipartite entanglement

Necessary and sufficient conditions for bipartite entanglement are derived, which apply to arbitrary Hilbert spaces. Motivated by the concept of witnesses, optimized entanglement inequalities are formulated solely in terms of arbitrary Hermitian operators, which makes them useful for applications in experiments. The needed optimization procedure is based on a separability eigenvalue problem, whose analytical solutions are derived for a special class of projection operators. For general Hermitian operators, a numerical implementation of entanglement tests is proposed. It is also shown how to identify bound entangled states with positive partial transposition.Comment: 7 pages, 2 figur

Sharing the Burden of Collective Security in the European Union. Research Note

This article compares European Union (EU) burden-sharing in security governance distinguishing between assurance, prevention, protection, and compellence policies. We employ joint-product models and examine the variation in the level of publicness, the asymmetry of the distribution of costs and benefits, and aggregation technologies in each policy domain. Joint-product models predict equal burden sharing for protection and assurance because of their respective weakest-link and summation aggregation technologies with symmetric costs. Prevention is also characterized by the technology of summation, but asymmetry of costs implies uneven burden-sharing. Uneven burden-sharing is predicted for compellence because it has the largest asymmetry of costs and a best-shot aggregation technology. Evaluating burden-sharing relative to a country?s ability to contribute, Kendall tau-tests examine the rank-correlation between security burden and the capacity of EU member states. These tests show that the smaller EU members disproportionately shoulder the costs of assurance and protection; wealthier EU members carry a somewhat disproportionate burden in the provision of prevention, and larger EU members in the provision of compellence. When analyzing contributions relative to expected benefits, asymmetric marginal costs can largely explain uneven burden-sharing. The main conclusion is that the aggregated burden of collective security governance in the EU is shared quite evenly

Convex ordering and quantification of quantumness

The characterization of physical systems requires a comprehensive understanding of quantum effects. One aspect is a proper quantification of the strength of such quantum phenomena. Here, a general convex ordering of quantum states will be introduced which is based on the algebraic definition of classical states. This definition resolves the ambiguity of the quantumness quantification using topological distance measures. Classical operations on quantum states will be considered to further generalize the ordering prescription. Our technique can be used for a natural and unambiguous quantification of general quantum properties whose classical reference has a convex structure. We apply this method to typical scenarios in quantum optics and quantum information theory to study measures which are based on the fundamental quantum superposition principle.Comment: 9 pages, 2 figures, revised version; published in special issue "150 years of Margarita and Vladimir Man'ko