We consider the problem of obtaining the entanglement witnesses for a given quantum system. We present a method to convert this problem to a standard convex optimization problem by defining a feasible region. Then we develop a generic two-step algorithm for this problem which can be applied to the entanglement detection of $N$-partite quantum system. This permits an organized construction of nonlinear and linear entanglement witnesses for a given quantum system. To demonstrate the capability of this approach, we apply our results to several three-qubit systems which in some cases lead to necessary and sufficient conditions for separability.
We consider the problem of obtaining the entanglement witnesses for a given quantum system. We present a method to convert this problem to a standard convex optimization problem by defining a feasible region. Then we develop a generic two-step algorithm for this problem which can be applied to the entanglement detection of N-partite quantum system. This permits an organized construction of nonlinear and linear entanglement witnesses for a given quantum system. To demonstrate the capability of this approach, we apply our results to several three-qubit systems which in some cases lead to necessary and sufficient conditions for separability.
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
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
Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5.
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.
We focus on determining the separability of an unknown bipartite quantum state $\ensuremath{\rho}$ by invoking a sufficiently large subset of all possible entanglement witnesses given the expected value of each element of a set of mutually orthogonal observables. We review the concept of an entanglement witness from the geometrical point of view and use this geometry to show that the set of separable states is not a polytope and to characterize the class of entanglement witnesses (observables) that detect entangled states on opposite sides of the set of separable states. All this serves to motivate a classical algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of observables. The idea of such an algorithm, which is an efficient reduction of the quantum separability problem to a global optimization problem, was introduced by [Ioannou et al., Phys. Rev. A 70, 060303(R)], where it was shown to be an improvement on the naive approach for the quantum separability problem (exhaustive search for a decomposition of the given state into a convex combination of separable states). The last section of the paper discusses in more generality such algorithms, which, in our case, assume a subroutine that computes the global maximum of a real function of several variables. Despite this, we anticipate that such algorithms will perform sufficiently well on small instances that they will render a feasible test for separability in some cases of interest (e.g., in $3\ifmmode\times\else\texttimes\fi{}3$ dimensional systems).
A class of entanglement witnesses (EWs) called reduction-type entanglement witnesses is introduced, which can detect some multipartite entangled states including positive partial transpose ones with Hilbert space of dimension ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$. In fact the feasible regions of these EWs turn out to be convex polygons and hence the manipulation of them reduces to linear programming which can be solved exactly by using the simplex method. The decomposability and nondecomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamio\l{}kowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Hall [Phys. Rev. A 72, 022311 (2005)] are obtained.