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arXiv:1006.1594 · Physical Review A

Multiparticle covariance matrices and the impossibility of detecting graph-state entanglement with two-particle correlations

Oleg Gittsovich, Philipp Hyllus, Otfried Gühne

Published Sep 9, 2010• 53 citations
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Abstract

We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows detection of bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the Greenberger-Horne-Zeilinger (GHZ) states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states to which the GHZ states belong.

DOI
10.1103/physreva.82.032306

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We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows detection of bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the Greenberger-Horne-Zeilinger (GHZ) states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states to which the GHZ states belong.
We present a method to derive separability criteria for different classes of multiparticle entanglement, especially genuine multiparticle entanglement. The resulting criteria are necessary and sufficient for certain families of states. This, for example, completely solves the problem of classifying N -qubit Greenberger–Horne–Zeilinger states mixed with white noise according to their separability and entanglement properties. Further, the criteria are superior to all known entanglement criteria for many other families; also they allow the detection of bound entanglement. We next demonstrate that they are easily implementable in experiments and discuss applications to the decoherence of multiparticle entangled states.
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We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow us to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria--such as the computable cross-norm and local uncertainty criteria--as well as their limitations.
In this thesis, we discuss several aspects of the characterization of entanglement in multipartite quantum systems, including detection, classification and quantification of entanglement. First, we discuss triqubit pure entanglement and propose a special true tripartite entanglement, the mixed entanglement, besides the Greenberger-Horne-Zeilinger (GHZ) entanglement and the W entanglement. Then, based on quantitative complementarity relations, we draw entanglement Venn diagrams for triqubit pure states with different entanglements and introduce the total tangle Tau sup(T) to quantify total entanglement of triqubit pure states by defining the union I that is equivalent to the total tangle Tau sup(T) from the mathematical point of view. The generalizations of entanglement Venn diagrams and the union I to N-qubit pure states are also discussed. Finally, based on the ranks of reduced density matrices, we discuss the separability of multiparticle arbitrary -dimensional pure and mixed states, respectively.
We derive a family of necessary separability criteria for finite-dimensional systems based on inequalities for variances of observables. We show that every pure bipartite entangled state violates some of these inequalities. Furthermore, a family of bound entangled states and true multipartite entangled states can be detected. The inequalities also allow us to distinguish between different classes of true tripartite entanglement for qubits. We formulate an equivalent criterion in terms of covariance matrices. This allows us to apply criteria known from the regime of continuous variables to finite-dimensional systems.