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# Hermitian Tensor Product Approximation of Complex Matrices and Separability

Reports on Mathematical Physics (Impact Factor: 1.04). 04/2006; 57(2):271-288. DOI: 10.1016/S0034-4877(06)80021-2

Source: arXiv

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Naihuan Jing, Dec 30, 2013 Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

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**ABSTRACT:**This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types. - [Show abstract] [Hide abstract]

**ABSTRACT:**The dynamics of a central qubit system coupled to an isotropic ferromagnetic Lipkin–Meshkov–Glick spin bath at nonzero temperature is studied. We derive exactly the reduced density matrix and investigate the pairwise thermal entanglement, the coherence and the concurrence of the central spins. Through the behavior of these quantities, we show that at very low temperatures the dynamics is sensitive to the presence of the critical point of the environment.Journal of Physics A Mathematical and Theoretical 03/2008; 41(13):135302. DOI:10.1088/1751-8113/41/13/135302 · 1.69 Impact Factor