One of the important approaches to detect quantum entanglement is using linear entanglement witnesses (EW s). In this paper, by determining the envelope of the boundary hyper-planes defined by a family of linear EW s, a set of powerful nonlinear optimal EW s is manipulated. These EW s enable us to detect some three qubits bound MUB (mutually unbiased bases) diagonal entangled states, i.e., the PPT (positive partial transpose) entangled states. Also, in some particular cases, the introduced nonlinear optimal EW s are powerful enough to separate the bound entangled regions from the separable ones.Finally, we present numerical examples to demonstrate the practical accessibility of this approach.

Here we consider a class of 2 ⊗ 2 ⊗ d chessboard density matrices starting with threequbit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. 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 cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically.

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 * non-linear entanglement witnesses 2 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. Entanglement is one of the interesting features of quantum systems. It is used as a physical resource in realization of many quantum information and quantum computation processes such as quantum parallelism [1], quantum cryptography [2], quantum teleportation [3, 4], quantum dense coding [5, 6], reduction of communication complexity [7] and beating classical communication complexity bounds with entanglement [8].In these applications usually a source produces entangled particles and after these particles reach to the related parties, there is an important question for the parties -are these particles already entangled?One approach to distinguish entangled states from separable ones is entanglement witness (EW) [8,9]. A quantum state is entangled iff there exists a Hermitian operator W with T r(W ρ) < 0 and T r(W ρ sep ) 0 for any separable state ρ sep [10]. We say that the witness W detects the entanglement of density matrix. Recently there has been an increased interest in the nonlinear EWs because of their improved detection with respect to linear EWs. A nonlinear EW is any bound on nonlinear function of observables which is satisfied by separable states but violated by some entangled states [8,11,12,13]. non-linear entanglement witnesses

Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartite entanglement of mixed states. Nonlinear entanglement witnesses have the advantage of a wider detection range in the entangled region. We construct genuine entanglement witnesses for four qubits density matrices in the mutually unbiased basis. First, we find the convex feasible region with positive partial transpose states. Then, to reveal the entangled regions, we present some appropriate linear entanglement witnesses and, we find the envelope of this family of linear witnesses as a nonlinear witness. Finally, we study thermal entanglement and we show for some Hamiltonians the witnesses can detect the entanglement at all temperatures.

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