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arXiv:quant-ph/0103038 · Linear Algebra and its Applications

Convexity and the separability problem of quantum mechanical density matrices

A. O. Pittenger, Morton H. Rubin

Published May 1, 2002• 41 citations
arXivHTMLPDF
DOI
10.1016/s0024-3795(01)00524-9

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Here we consider a class of $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}d$ density matrices which have positive partial transposes with respect to all subsystems. The entanglement witness approach is used to investigate the entanglement of these density matrices. To demonstrate the approach, the three-qubit case is considered in detail. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider the cases in which the exact geometrical shape of the feasible region can be obtained. In this way, various linear and nonlinear EWs are constructed. The optimality and decomposability of some of the introduced EWs are also considered. Furthermore, the detection of the density matrices by the introduced EWs are discussed analytically and numerically.
The detection and quantification of quantum entanglement are central problems in quantum information science. Based on the realignment of density matrices and the vectorization of reduced density matrices, we construct a family of enhanced separability criteria for bipartite and multipartite quantum systems. Our criteria recover the previously proposed realignment-based criteria as special cases and detect more entanglement than the earlier ones. Moreover, we derive tighter lower bounds for the concurrence and the convex-roof extended negativity, criteria for detecting the genuine tripartite entanglement, and lower bounds on genuine-multipartite-entanglement concurrence. The advantages of our results are demonstrated with detailed examples.
A generic algorithm is developed to reduce the problem of obtaining linear and nonlinear entanglement witnesses of a given quantum system, to convex optimization problem. This approach is completely general and can be applied for the entanglement detection of any N-partite quantum system. For this purpose, a map from convex space of separable density matrices to a convex region called feasible region is defined, where by using exact convex optimization method, the linear entanglement witnesses can be obtained from polygonal shape feasible regions, while for curved shape feasible regions, envelope of the family of linear entanglement witnesses can be considered as nonlinear entanglement witnesses. This method proposes a new methodological framework within which most of previous EWs can be studied. To conclude and in order to demonstrate the capability of the proposed approach, besides providing some nonlinear witnesses for entanglement detection of density matrices in unextendible product bases, W-states, and GHZ with W-states, some further examples of three qubits systems and their classification and entanglement detection are included. Also it is explained how one can manipulate most of the non-decomposable linear and nonlinear three qubits entanglement witnesses appearing in some of the papers published by us and other authors, by the method proposed in this paper. Keywords: non-linear and linear entanglement witnesses, convex optimization. PACS number(s): 03.67.Mn, 03.65.Ud