Characterization of four-qubit states via Bell inequalities

Abstract

A set of Bell inequalities classifying the quantum entanglement of four-qubit states is presented. These inequalities involve only two measurement settings per observer and can characterize fully separable, bi-separable and tri-separable quantum states. In addition, a quadratic inequality of the Bell operators for four-qubit systems is derived. Bell inequalities, separability, Bell operatorsCitation: Zhao H, Zhang X H, Fei S M, et al. Characterization of four-qubit states via Bell inequalities. Chin Sci Bull, 2013Bull, , 58: 2334Bull, -2339Bull, , doi: 10.1007 The Bell inequality [1] provided the first possibility to distinguish experimentally between quantum-mechanical predictions and those of local realistic models. Derivations of new and stronger Bell inequalities are one of the most important and challenging subjects in quantum information processing. Since Bell's work, there were many important generalizations such as [2-11] and references therein. The Bell inequalities presented in [12] involve only two measurement settings per observer and can detect perfectly the quantum entanglement of the generalized GHZ states. By using the idea in constructing Bell operators [12], a set of new Bell inequalities are given in [13], which gives rise to a finer classification of the entanglement for three-qubit systems.The entanglement of four-qubit systems has been treated in terms of Bell inequalities of Mermin-Klyshko type. In [14] the quantum nonlocality of some four-qubit states, the GHZ state, W state, cluster state and the state proposed in [15], has been investigated, towards the optimal violations of the Bell inequality for these states. The classification of entanglement has been also studied in such as [16][17][18][19] with linear inequalities for qubit systems and [20] with non-linear inequalities for detecting bi-separable states in arbitrary dimensional quantum systems.In this work, we study the quantum entanglement of four-*Corresponding author (email: zhaohui@bjut.edu.cn) qubit systems by using the idea in constructing Bell operators in [12]. We generalize the results of three-qubit systems in [13] to four-qubit systems. It has been shown that the standard Werner-Wolf-Żukowski-Brukner (WWZB) inequalities cannot detect the entanglement of the generalized Greenberger-Horne-Zeilinger (GHZ) states given by |ψ = cos α|0, ..., 0 + sin α|1, ..., 1 with 0 α π/4 [21, 22]. However, the Bell operators constructed in the way provided in [12] can detect the entanglement of the generalized GHZ state wholly. Our Bell operators are constructed by using the idea in [12]. The resulted Bell inequalities can distinguish fully separable, bi-separable and tri-separable states of a four-qubit system. Moreover, these linear Bell inequalities involve only two measurement settings per observer. Analytical formulas of the average values of the Bell operators for four-qubit systems are also derived. And a quadratic inequality of the Bell operators for all four-qubit systems has been presented. Explicit geometrical pictures show t...

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