Entanglement witnesses: construction, analysis and classification

Abstract

From the physical point of view entanglement witnesses define a universal\ntool for analysis and classification of quantum entangled states. From the\nmathematical point of view they provide highly nontrivial generalization of\npositive operators and they find elegant correspondence with the theory of\npositive maps in matrix algebras. We concentrate on theoretical analysis of\nvarious important notions like (in)decomposability, atomicity, optimality,\nextremality and exposedness. Several methods of construction are provided as\nwell. Our discussion is illustrated by many examples enabling the reader to see\nthe intricate structure of these objects. It is shown that the theory of\nentanglement witnesses finds elegant geometric formulation in terms of convex\ncones and related geometric structures.\n