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arXiv:0904.3860 · Physical Review Letters

Multipartite Entanglement Detection via Structure Factors

Philipp Krammer, Hermann Kampermann, Dagmar Bruß, Reinhold A. Bertlmann, L. C. Kwek, Chiara Macchiavello

Published Sep 3, 2009• 77 citations
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Abstract

We establish a relation between entanglement of a many-body system and its diffractive properties, where the link is given by structure factors. Based on these, we provide a general analytical construction of multiqubit entanglement witnesses. The proposed witnesses contain two-point correlations. They could be either measured in a scattering experiment or via local measurements, depending on the underlying physical system. For some explicit examples of witnesses we analyze the properties of the states that are detected by them. We further study the robustness of these witnesses with respect to noise.

DOI
10.1103/physrevlett.103.100502

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We establish a relation between entanglement of a many-body system and its diffractive properties, where the link is given by structure factors. Based on these, we provide a general analytical construction of multiqubit entanglement witnesses. The proposed witnesses contain two-point correlations. They could be either measured in a scattering experiment or via local measurements, depending on the underlying physical system. For some explicit examples of witnesses we analyze the properties of the states that are detected by them. We further study the robustness of these witnesses with respect to noise.
Multi-qubit entanglement could act as a useful resource for many quantum information communication and processing tasks like quantum networking processes. Genuine non-trivial multi-qubit entanglement could delineates quantum physics from classical physics. Detection of multipartite entanglement however is a theoretical and experimental challenge. A useful tool for this purpose are entanglement witnesses. Recently, we provide an analytical construction of multi-qubit entanglement witnesses using static structure factors (SSF) and study the robustness of the witnesses with respect to noise. In this article, we consider the effectiveness of the entanglement witness based on SSF for probing spin chain.
This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.
We construct an entanglement witness for many-qubit systems, based on symmetric two-body correlations with two measurement settings. This witness is able to detect the entanglement of some Dicke states for any number of particles and such detection exhibits some robustness against white noise and thermal noise under the Lipkin-Meshkov-Glick Hamiltonian. In addition, it detects the entanglement of spin-squeezed states, with a detection strength that approaches the maximal value for sufficiently large numbers of particles. As spin-squeezed states can be experimentally generated, the properties of the witness with respect to these states may be amenable to experimental investigation. We show that while the witness is unable to detect Greenberger-Horne-Zeilinger (GHZ) states, it is instead able to detect superpositions of Dicke states with GHZ states.
We present entanglement witness operators for detecting genuine multipartite entanglement. These witnesses are robust against noise and require only two local measurement settings when used in an experiment, independent of the number of qubits. This allows detection of entanglement for an increasing number of parties without a corresponding increase in effort. The witnesses presented detect states close to Greenberger-Horne-Zeilinger, cluster, and graph states. Connections to Bell inequalities are also discussed.
Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Because of the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this Letter, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have an effective implementation with either the generalized control-swap quantum circuits or the random unitary techniques. As an example, in the bipartite scenario, we develop an entanglement criterion that can detect bound entanglement and show strong detection capability in the multiqubit Ising model with a long-range XY Hamiltonian. In the multipartite case, the permutation-moment-based criteria can detect entangled states that are not detectable by any criteria extended from the bipartite case. Our framework also shows potential in entanglement quantification and entanglement structure detection.