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We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness multipartite entanglement of 10 3 coupled quantum oscillators, by solving our basic equations analytically. This clearly demonstrates the feasibility of our method for studying ultrahigh orders of multipartite entanglement in complex quantum systems.PACS numbers: 03.67. Mn, 03.65.Ud, 42.50.Dv Entanglement represents a fundamental quantum correlation between compound quantum systems. Since the early days of quantum physics this property has been used to illustrate the surprising consequences of the quantum description of nature [1,2]. Moreover, entanglement plays a fundamental role in various applications and protocols in quantum information science [3][4][5].In multipartite systems a separable state is a statistical mixture of product states [6]. A quantum state is entangled, whenever it cannot be represented in this form. Various forms of multipartite entanglement are known [7][8][9][10]. The most prominent and nonequivalent forms of entangled multipartite quantum states are the GHZ-state [11] and the W-state [12], which have been generalized to so-called cluster and graph states [13,14]. Another classification is given in terms of partial and full (or genuine) multipartite entanglement, for an introduction see e.g. [4,5]. Beyond finite dimensional systems, multipartite quantum entanglement in continuous variable systems turns out to be a cumbersome problem. Even in the case of Gaussian states, there exist multipartite entangled states, which cannot be distilled [15].High orders of multipartite entanglement are of great interest, for example, in quantum metrology. Multipartite entanglement has been shown to be essential to reach the maximal sensitivity in metrological tasks [26]. In this context, the quantum Fisher information has been used to characterize the entanglement [27][28][29].The detection of entanglement is typically done via the construction of proper entanglement witnesses [16][17][18], being equivalent to the method of positive, but not completely positive maps. A witness is an observable, which is non-negative for separable states, and it can have a negative expectation value for entangled states. For different kinds of entanglement, different types of witnesses have been considered: bipartite witnesses [17,19]; Schmidt number witnesses [20,21]; and multipartite witnesses for partial and genuine entanglement [22][23][24][25]. A systematic approach for witnessing entanglement in complex quantum systems is missing yet.Recently, we considered the construction of bipartite entanglement witnesses with the so-called separability eigenvalue equations [19]. We have shown that the same equations need to be solved to obtain entanglement quasiprobabilities, which are nonpositive distributions if and only if the corresponding quantum state is entangled [30]. Moreover, we have s...

We provide necessary and sufficient conditions for the partial transposition of bipartite harmonic quantum states to be nonnegative. The conditions are formulated as an infinite series of inequalities for the moments of the state under study. The violation of any inequality of this series is a sufficient condition for entanglement. Previously known entanglement conditions are shown to be special cases of our approach.PACS numbers: 03.67. Mn, 03.65.Ud, 42.50.Dv Entanglement plays a key role in the rapidly developing field of quantum information processing. In this context it is important to provide methods for characterizing entangled quantum states on the basis of observable quantities. However, already in seemingly simple cases this problem turns out to be rather complex. Even for a two-party harmonic-oscillator system so far there exists no complete characterization of entanglement to be used in experiments.For characterizing entanglement, that is inseparability, of the density operator of a bipartite continuous variable system, one may use the Peres-Horodecki condition [1,2,3]. A sufficient condition for entanglement consists in the negativity of the partial transposition (NPT) of the quantum state of the two-party system. To characterize NPT for such a system completely, however, to our best knowledge is still an unsolved problem.A sufficient condition for the NPT has been proposed by Simon [4]. It is based on second-order moments of position and momentum operators. For the special case of Gaussian states the resulting entanglement criterion has been shown to be necessary and sufficient. Another inseparability condition based on second moments has been derived without explicitely using the NPT condition [5]. This condition is also complete for the characterization of entanglement of two-mode Gaussian states. The latter approach has been extended to special higher-order moments [6,7] and even to more general operator functions [8].The complexity of the problem under study may become clear when we go back to a related but simpler problem. It consists in the characterization of nonclassical effects based on the negativity of the Glauber-Sudarshan P -function. Only recently the problem was solved of how to characterize the nonclassicality in terms of observable quantities. This requires an infinite hierarchy of conditions formulated either in terms of characteristic functions [9] or in terms of observable moments [10,11].In the present contribution we will further develop the concept of the complete characterization of single-mode nonclassicality with the aim to characterize the entanglement of bipartite continuous variable quantum states. Based on the NPT condition we derive a hierarchy of necessary and sufficient conditions for the NPT in terms of observable moments. Even though this only leads to sufficient conditions for entanglement, it can be applied to a variety of quantum states. It does not only contain as a special case the condition of Simon [4], but also other types of inequalities [5,6,7,8]. For ...

The nonclassicality of single-mode quantum states is studied in relation to the entanglement created by a beam splitter. It is shown that properly defined quantifications -based on the quantum superposition principle -of the amounts of nonclassicality and entanglement are strictly related to each other. This can be generalized to the amount of genuine multipartite entanglement, created from a nonclassical state by an N splitter. As a consequence, a single-mode state of a given amount of nonclassicality is fully equivalent, as a resource, to exactly the same amount of entanglement. This relation is also considered in the context of multipartite entanglement and multimode nonclassicality.