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arXiv:quant-ph/0011025 · Journal of Physics A Mathematical and General

Multiparticle entanglement and its experimental detection

Wolfgang Dür, J. I. Cirac

Published Aug 24, 2001• 44 citations
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Abstract

We discuss several aspects of multiparticle mixed-state entanglement and its experimental detection. First we consider entanglement between two particles which is robust against disposals of other particles. To completely detect these kinds of entanglement, full knowledge of the multiparticle density matrix (or of all reduced density matrices) is required. Then we review the relation of the separability properties of l -partite splittings of a state ρ to its multipartite entanglement properties. We show that it suffices to determine the diagonal matrix elements of ρ in a certain basis in order to detect multiparticle entanglement properties of ρ. We apply these observations to analyse two recent experiments, where multiparticle entangled states of 3 (4) particles were produced. Finally, we focus on bound entangled states (non-separable, non-distillable states) and show that they can be activated by joint actions of the parties. We also provide several examples which show the activation of bound entanglement with bound entanglement.

DOI
10.1088/0305-4470/34/35/310

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We discuss several aspects of multiparticle mixed-state entanglement and its experimental detection. First we consider entanglement between two particles which is robust against disposals of other particles. To completely detect these kinds of entanglement, full knowledge of the multiparticle density matrix (or of all reduced density matrices) is required. Then we review the relation of the separability properties of l -partite splittings of a state ρ to its multipartite entanglement properties. We show that it suffices to determine the diagonal matrix elements of ρ in a certain basis in order to detect multiparticle entanglement properties of ρ. We apply these observations to analyse two recent experiments, where multiparticle entangled states of 3 (4) particles were produced. Finally, we focus on bound entangled states (non-separable, non-distillable states) and show that they can be activated by joint actions of the parties. We also provide several examples which show the activation of bound entanglement with bound entanglement.
We analyze the existence of activable bound entangled states in multiparticle systems. We first give a series of examples that illustrate some different ways in which bound entangled states can be activated by letting some of the parties share maximally entangled states. Then, we derive necessary conditions for a state to be distillable as well as to be activable. These conditions turn out to be also sufficient for a certain family of multiqubit states. We use these results to explicitely construct states displaying properties related to bound entanglement and its activation.
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 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.
Since the 1980s, there has been a physical problem relating to the detection of multipartite entangled states that helps us to investigate fundamental point of view on interpretations of quantum mechanics. On the other hand, information theories based on the mathematical structure of quantum mechanics, for example, quantum cryptography and quantum computation, have been investigated by researchers whose number has been increasing day by day in the world. This field became to be called quantum information theories. A practical experimental realization of quantum information processing needs that we detect experimentally multipartite entangled states. The number of researchers for multipartite entangled states is increasing rapidly. Especially, Greenberger-Horne-Zeilinger (GHZ) states are useful for quantum information processing information processing using quantum mechanics, which improves the conventional information processing ability which had been based on classical theories. However, practical experimental realization of the theories and controlling quantum states have inevitable difficulties due to the interaction with environments. The requirement in maintaining multipartite entangled states in relation to the time that takes to make a practical experiment is also a problem. Many experimentalists are tackling this problem, and recently, several groups have reported that multipartite entangled states have been physically detected in their laboratories. The main difficulty is how to judge whether or not the observed quantum state is indeed a multipartite entangled state. For this purpose, it is necessary to construct theoretical models to analyze such experimental data. In this thesis, we apply a mathematical structure as to density operators described in a high-dimensional Hilbert space, and we employ mathematical theories to prove if the corresponding density operator represents a multipartite entangled state or not. The main aim for constructing such models is to analyze the raw experimental data. Moreover, this model helps experimentalists in the sense that it allows them to see how to confirm that observed states are indeed the desired state. With mathematical theories developed before this thesis, only limited applications for this purpose were allowed. Some theories cover two-dimensional Hilbert space only. It has not been even discussed how to analyze the fidelity to GHZ states of the observed state. This has led to some doubt of that experimental data might not always confirm three-partite entangled states. However, by using the method of this thesis, statistical check based on such experimental data of the existence of three particle-entangled states is possible under appropriate assumptions. This example symbolizes that our analytical knowledge has improved very much in the last few years. This thesis gives us an advanced mathematical model for analyzing experimental data.
We investigate the entanglement properties of multiparticle systems, concentrating on the case where the entanglement is robust against the disposal of particles. Two qubits---belonging to a multipartite system---are entangled in this sense if their reduced density matrix is entangled. We introduce a family of multiqubit states, for which one can choose for any pair of qubits independently whether they should be entangled or not as well as the relative strength of the entanglement, thus providing the possibility to construct all kinds of ``entanglement molecules.'' For some particular configurations, we also give the maximal amount of entanglement achievable.