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Skowronek L. Dualities and positivity in the study of quantum entanglement
Inteational Joual of Quantum Information, Vol. 8, No. 5 (2010) 721-754
by World Scientific Publishing Company

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
speci?c 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 su±cient, which should be useful in case of future e®orts in that direction.
Finally, we introduce the concept of length of a separable state and present new results
conceing relationships between the length and Schmidt rank. In particular, we prove that
separable states of length lower or equal to 3 have Schmidt ranks equal to their lengths. We also
give an example of a state which has length 4 and Schmidt rank 3.
Keywords: Quantum entanglement; separable sates; entanglement witnesses; positive maps;
convex cones.
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