2009/02/27 by Łukasz Skowronek, Lukasz Skowronek, Erling Størmer +3 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Block (permutation group theory) #Duality (order theory) #Hilbert space #Isomorphism (crystallography) #Quantum Information and Cryptography #Quantum entanglement #Simple (philosophy) #Space (punctuation) #math.OA #quant-ph
paper · pdf · doi:10.1063/1.3155378
published as J. Math. Phys. 50, 062106 (2009) · 22 pages, 3 figures
arxiv created 2009/02/27 · openalex publication_date 2009/06/01 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiołkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.