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Unentangled pure states on a bipartite system are exactly the coherent stateswith respect to the group of local transformations. What aspects of the studyof entanglement are applicable to generalized coherent states? Conversely, whatcan be learned about entanglement from the well-studied theory of coherentstates? With these questions in mind, we characterize unentangled pure statesas extremal states when considered as linear functionals on the local Liealgebra. As a result, a relativized notion of purity emerges, showing thatthere is a close relationship between purity, coherence and (non-)entanglement.To a large extent, these concepts can be defined and studied in the even moregeneral setting of convex cones of states. Based on the idea that entanglementis relative, we suggest considering these notions in the context of partiallyordered families of Lie algebras or convex cones, such as those that arisenaturally for multipartite systems. The study of entanglement includes notionsof local operations and, for information-theoretic purposes, entanglementmeasures and ways of scaling systems to enable asymptotic developments. Wepropose ways in which these may be generalized to the Lie-algebraic setting,and to a lesser extent to the convex-cones setting. One of our originalmotivations for this program is to understand the role of entanglement-likeconcepts in condensed matter. We discuss how our work provides tools foranalyzing the correlations involved in quantum phase transitions and otheraspects of condensed-matter systems.Comment: 37 page

Generalizations of entanglement based on coherent states and convex sets