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arXiv:0911.2307

Quantifying entanglement of two relativistic particles via decomposable optimal entanglement witnesses

M. A. Jafarizadeh, M. Mahdian

Published Nov 12, 2009• 0 citations
arXivHTMLPDF

Abstract

The study of Entanglement of Formation of a mixed state of a bipartite system in high-dimensional Hilbert space is not easy in general. So, we focus on determining the amount of entanglement for a bipartite mixed state based on the concept of decomposable optimal entanglement witness (DOEW), that can be calculated as a minimum distance of an entangled state from the edge of positive partial transpose (PPT) states which has the most negative (positive) expectation value for non-PPT (bound) entangled states. We have constructed DOEWs based on the convex optimization method, then by using of it we quantify the degree of entanglement for two spin half particles under the Lorentz transformations. For convenience, we restrict ourselves to 2D momentum subspace and under this constraint when the momentum and the Lorentz boost are parallel, we have shown that the entanglement is not relativistic invariant. Keywords : Relativistic entanglement, Measure of entanglement, Optimal entanglement witnesses, Convex optimization PACS numbers: 03.67.Hk, 03.65.Ta

DOI
10.48550/arxiv.0911.2307

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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.
We develop an approach of quantifying entanglement in mixed quantum states by the optimal entanglement witness operator. We identify the convex set of mixed states for which a single witness provides the exact value of an entanglement measure and show that the convexity, properties, and symmetries of entanglement or of a target state considerably fix the form of the optimal witness. This greatly reduces the difficulty in computing and experimentally determining entanglement measures. As an example, we show how to experimentally quantify bound entanglement in four-qubit noisy Smolin states and three-qubit Greenberger-Horne-Zeilinger entanglement under white noise. For general measures and states, we provide a numerical method to efficiently optimize the witness.
Abstract An entanglement witness (EW) is a Hermitian operator that can distinguish an entangled state from all separable states. We developed and implemented a numerical method based on machine learning to create multipartite EW. Using a Support Vector Machine (SVM) algorithm, we construct EW's based on local orthogonal observables in the form of a hyperplane that separates the separable region from the entangled state for two, three, and four qubits Bell-diagonal mixed states which can be generalized to a multipartite mixed state. One of the important features of this method is that the EW s are optimal and are completely tangent to the separable region. Also, we generate non-decomposable EW's that can detect positive partial transpose entangled states (PPTES).
We develop a strong and computationally simple entanglement criterion. The criterion is based on an elementary positive map Phi which operates on state spaces with even dimension N > or = 4. It is shown that Phi detects many entangled states with a positive partial transposition (PPT) and that it leads to a class of optimal entanglement witnesses. This implies that there are no other witnesses which can detect more entangled PPT states. The map Phi yields a systematic method for the explicit construction of high-dimensional manifolds of bound entangled states.
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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