How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given to the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.

Entanglement, its generation, manipulation and fundamental understanding is at the very heart of quantum mechanics. The phrase entanglement was coined by Erwin Schrödinger in 1935 for particles that are described by a common wave function where individual particles are not independent of each other but where their quantum properties are inextricably interwoven 1 . Entanglement properties of two and three particles have been studied extensively and are very well understood. Entanglement of four 2 and five 3 particles was demonstrated experimentally. However, both creation and characterization of entanglement become exceedingly difficult for multi-particle systems. Thus the availability of such multiparticle entangled states together with the full information on these states in form of their 1

Graph states are special kinds of multipartite entangled states that correspond to mathematical graphs where the vertices take the role of quantum spin systems and the edges represent interactions. They not only provide an efficient model to study multiparticle entanglement, but also find wide applications in quantum error correction, multi-party quantum communication and most prominently, serve as the central resource in one-way quantum computation. Here we report the creation of two special instances of graph states, the six-photon Greenberger-Horne-Zeilinger states -- the largest photonic Schr\"{o}dinger cat, and the six-photon cluster states-- a state-of-the-art one-way quantum computer. Flexibly, slight modifications of our method allow creation of many other graph states. Thus we have demonstrated the ability of entangling six photons and engineering multiqubit graph states, and created a test-bed for investigations of one-way quantum computation and studies of multiparticle entanglement as well as foundational issues such as nonlocality and decoherence

We present the experimental detection of genuine multipartite entanglement using entanglement witness operators. To this aim we introduce a canonical way of constructing and decomposing witness operators so that they can be directly implemented with present technology. We apply this method to three-and four-qubit entangled states of polarized photons, giving experimental evidence that the considered states contain true multipartite entanglement. [3,4] as it gives a simple sufficient and necessary condition for entanglement. Yet, the situation is much more complicated for higher dimensional and multipartite systems, where simple necessary and sufficient conditions are not known [5].In the analysis of multipartite systems, it is important to distinguish between genuine multipartite entanglement and biseparable (triseparable, etc.) entanglement. Genuine multipartite entangled pure states cannot be created without participation of all parties. Conversely, for pure biseparable states of n parties a group of m < n parties can be found which are entangled among each other, but not with any member of the other group of n − m parties [6]. Distinction and detection of genuine multipartite entanglement poses an important challenge in quantum information science. Bell inequalities are not suited to this aim in general. Multiseparable and biseparable states violate known Bell inequalities less than npartite Greenberger-Horne-Zeilinger (GHZ) states. However, for n > 3 there exist even pure n-partite entangled states with a lower violation than biseparable states [7]. Only recently, significant progress in classifying multipartite entanglement has been achieved using entanglement witnesses [4,8]. These observables can always be used to detect various forms of multipartite entanglement, when some a priori knowledge about the states under investigation is provided [9]; they are in this sense more powerful than Bell inequalities.A witness of genuine n-partite entanglement is an observable which has a positive expectation value on states with n − 1 partite entanglement and a negative expectation value on some n-partite entangled states. The latter states and their entanglement, respectively, are said to be detected by W. Witnesses provide sufficient criteria for entanglement and for distinguishing the various classes of genuine multipartite entangled states.The goal of this Letter is twofold. First, we introduce a general scheme for the construction of multipartite witness operators and their decomposition into locally measurable observables. In this way, we demonstrate how witness operators can be implemented experimentally in a straightforward way by using local projective measurements, even for multipartite systems [10]. Then, we apply this scheme to certain states and perform the experimental detection of their multipartite entanglement, which could not be revealed by known Bell inequalities. In particular, we use this method for the characterization of the three-qubit W state [11], and the four-qubit state |Ψ (4) [12]. A wit...

The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate [1]. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles [2]. Along this line, some types of hidden variable models have been experimentally ruled out [3,4,5,6,7,8,9]. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the results of Kochen, Specker, and Bell[10,11,12] show that non-contextuality is in conflict with quantum mechanics. The conflict resides in the structure of the theory and is independent of the properties of special states. It has been debated whether the Kochen-Specker theorem could be experimentally tested at all [13,14]. Only recently, first tests of quantum contextuality have been proposed and undertaken with photons [15] and neutrons [16,17]. Yet these tests required the generation of special quantum states and left various loopholes open. Here, using trapped ions, we experimentally demonstrate a state-independent conflict with non-contextuality. The experiment is not subject to the detection loophole and we show that, despite imperfections and possible measurement disturbances, our results cannot be explained in non-contextual terms. PACS numbers:Hidden variable models assert that the result v(A) of measuring the observable A on an individual quantum system is predetermined by a hidden variable λ. Two observables A and B are mutually compatible, if the result of A does not depend on whether B is measured before, after, or simultaneously with A and vice versa. Non-contextuality is the property of a hidden variable model that the value v(A) is determined, regardless of which other compatible observable is measured jointly with A. As a consequence, for compatible observables the relation v(AB) = v(A)v(B) holds. Kochen and Specker showed that the assumption of noncontextuality cannot be reconciled with quantum mechanics. A considerable simplification of the original Kochen-Specker argument by Mermin and Peres [18,19] uses a 3 × 3 square of observables A ij with possible outcomes v(A ij ) = ±1, where the observables in each row or column are mutually compatible. Considering the products of rows, the total product would be k=1,2,3 R k C k = 1, since any v(A ij ) appears twice in the total product.In quantum mechanics, however, one can take a fourlevel quantum system, for instance two spin-1 2 -particles, * Electronic address: christian.roos@uibk.ac.at and the following array of observables,(1) Here, σ (k) i denotes the Pauli matrix acting on the k-th particle, and all the observables have the outcomes ±1. Moreover, in each of the rows or columns of (1), the observables are mutually commuting and can be measured simultaneously or in any order. In any row or column, their measurement product R k or C k equa...

We present an approach to characterize genuine multiparticle entanglement by using appropriate approximations in the space of quantum states. This leads to a criterion for entanglement which can easily be calculated by using semidefinite programing and improves all existing approaches significantly. Experimentally, it can also be evaluated when only some observables are measured. Furthermore, it results in a computable entanglement monotone for genuine multiparticle entanglement. Based on this, we develop an analytical approach for the entanglement detection in cluster states, leading to an exponential improvement compared with existing schemes.

We present a method to derive separability criteria for the 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.

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.