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arXiv:quant-ph/0507035 · Physical Review A

Bell-state diagonal-entanglement witnesses

M. A. Jafarizadeh, M. Rezaee, S. K. A. Seyed Yagoobi

Published Dec 6, 2005• 27 citations
arXivHTMLPDF

Abstract

It has been shown that finding generic Bell-state diagonal-entanglement witnesses for ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ systems reduces to linear programming if the feasible region is a polygon by itself, and it can be solved approximately via linear programming if the feasible region is encircled by a polygon. Since solving linear programming for the generic case is difficult, multiqubit, $2\ensuremath{\bigotimes}N$ and $3\ensuremath{\bigotimes}3$ systems for the special case of generic Bell-state diagonal-entanglement witnesses for some particular choice of parameters have been considered. We obtain the optimal nondecomposable entanglement witness for a $3\ensuremath{\bigotimes}3$ system for some particular choice of parameters. By proving the optimality of the well-known reduction map and combining it with the optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some $3\ensuremath{\bigotimes}3$ bound entangled states are so detected. So the well-known Choi map as a particular case of the positive map in connection with this witness via Jamiolkowski isomorphism has been considered.

DOI
10.1103/physreva.72.062106

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Following the linear programming prescription of Jafarizadeh et al. [Phys. Rev. A 72, 062106 (2005)], the $d\ensuremath{\bigotimes}d$ Bell diagonal entanglement witnesses are provided. By using Jamiolkowski isomorphism, it is shown that the corresponding positive maps are the generalized qudit Choi maps. Also by manipulating particular $d\ensuremath{\bigotimes}d$ Bell diagonal separable states and constructing corresponding bound entangled states, it is shown that the obtained $d\ensuremath{\bigotimes}d$ Bell states diagonal entanglement witnesses (consequently qudit Choi maps) are nondecomposable in a certain range of their parameters.
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.
Two kinds of Bell-state diagonal (BSD) entanglement witnesses (EWs) are constructed by using the algebra of Dirac $\ensuremath{\gamma}$ matrices in a space-time of arbitrary dimension ddd, where the first kind can detect some BSD relativistic and nonrelativistic mmm-partite multispinor bound entangled states in a Hilbert space of dimension ${2}^{m\ensuremath{\lfloor}d/2\ensuremath{\rfloor}}$, including the bipartite Bell-type and isoconcurrence-type states in four-dimensional space-time (d=4)(d=4)(d=4). By using the connection between the Hilbert-Schmidt measure and the optimal EWs associated with the states, it is shown that, as far as the spin quantum correlations are concerned, the amount of entanglement is not a relativistic scalar and has no invariant meaning. The introduced EWs are manipulated via linear programming (LP) and can be solved exactly by using the simplex method. The decomposability or nondecomposability of these EWs is investigated, the region of nondecomposable EWs of the first kind is partially determined and it is shown that, all of the EWs of the second kind are decomposable. These EWs have the property that in bipartite systems they can determine the region of separable states, i.e., bipartite nondetectable density matrices of the same type as the EWs of the first kind are necessarily separable. Also, multispinor EWs with nonpolygon feasible regions are provided, where the problem is solved by approximate LP, and in contrast to the exactly manipulatable EWs, both the first and second kinds of the optimal approximate EWs can detect some bound entangled states.
Two classes of Bell-diagonal indecomposable entanglement witnesses in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ are considered. Within the first class, we find a generalization of the well-known Choi witness from ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, while the second one contains the reduction map. Interestingly, contrary to the ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ case, the generalized Choi witnesses are no longer optimal. We perform an optimization procedure of finding spanning vectors that eventually gives rise to optimal witnesses. Operators from the second class turn out to be optimal, however, without the spanning property. This analysis sheds light onto the intricate structure of optimal entanglement witnesses.
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