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...

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.

We derive a closed photo-counting formula, including noise counts and a finite quantum efficiency, for photon number resolving detectors based on on/off detectors. It applies to detection schemes such as array detectors and multiplexing setups. The result renders it possible to compare the corresponding measured counting statistics with the true photon number statistics of arbitrary quantum states. The photo-counting formula is applied to the discrimination of photon numbers of Fock states, squeezed states, and odd coherent states. It is illustrated for coherent states that our formula is indispensable for the correct interpretation of quantum effects observed with such devices.

Any bipartite quantum state has quasiprobability representations in terms of separable states. For entangled states these quasiprobabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one may reconstruct such quasiprobabilities from experimental data. Because of ambiguity, the quasiprobabilities obtained by the bare reconstruction are insufficient to identify entanglement. An optimization procedure is introduced to derive quasiprobabilities with a minimal amount of negativity. Negativities of optimized quasiprobabilities are necessary and sufficient for entanglement; their positivity proves separability.

An analysis is conducted of the multipartite entanglement for Gaussian states generated by the parametric down-conversion of a femtosecond frequency comb. Using a recently introduced method for constructing optimal entanglement criteria, a family of tests is formulated for mode decompositions that extends beyond the traditional bipartition analyses. A numerical optimization over this family is performed to achieve maximal significance of entanglement verification. For experimentally prepared 4-, 6-, and 10-mode states, full entanglement is certified for all of the 14, 202, and 115 974 possible nontrivial partitions, respectively.