Dualities and positivity in the study of quantum entanglement - Quantum PhysicsReport as inadecuate

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Abstract: We present a survey on mathematical topics relating to separable states andentanglement witnesses. The convex cone duality between separable states andentanglement witnesses is discussed and later generalized to other families ofoperators, leading to their characterization via multiplicative properties. Thecondition for an operator to be an entanglement witness is rephrased as aproblem of positivity of a family of real polynomials. By solving the latter ina specific case of a three-parameter family of operators, we obtain explicitdescription of entanglement witnesses belonging to that family. A relatedproblem of block positivity over real numbers is discussed. We also consider abroad family of block positivity tests and prove that they can never besufficient, which should be useful in case of future efforts in that direction.Finally, we introduce the concept of length of a separable state and presentnew results concerning relationships between the length and Schmidt rank. Inparticular, we prove that separable states of length lower of equal 3 haveSchmidt ranks equal to their lengths. We also give an example of a state whichhas length 4 and Schmidt rank 3.

Author: Lukasz Skowronek

Source: https://arxiv.org/

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