Entanglement is an inherent consequence of multipartite quantum systems and as a physical resource, quantum entanglement has many important applications in quantum computation and quantum information.^[1] It is worth noting that a fundamental question in the theory of entanglement is how to characterize and quantify entanglement.^{[2– 9]} In recent years, the characterization for multipartite entangled states has been considerably developed. The classification of a bipartite system under local unitary operators can be carried out by singular value decomposition.^[1] Under stochastic local operations and classical communication (SLOCC), a three-qubit entangled pure state can be converted either to the GHZ-state or to the W-state.^{[10, 11]} In the case of a four-qubit entangled pure state, it has been shown^[12] that there are nine classes of states. Recently, for the pure multipartite entangled states, Liu et al.^[13] proposed a method for local unitary classification using the symmetry and tensor decomposition. For quantification of the entanglement, as is well-known, one can take the concurrence^[14] as a measure of bipartite entanglement. In 2000, Coffman et al.^[15] proposed the three-tangle which is a measure of genuine tripartite entanglement. Next the expressions of four-tangle^{[16– 18]} and n-tangle^{[19, 20]} have been derived.

Quantum theory allows correlations between spatially separated systems, while local hidden variable (LHV) theory gives an upper bound that quantifies the degree of nonlocality for a given inequality. A famous inequality as a way of verifying predictions of quantum mechanics versus those of all possible LHV theories was first proposed by Bell.^[21] In 1969, Clauser et al. derived another correlation inequality, the Clauser– Horne– Shimony– Holt (CHSH) inequality, ^[22] which also provides a way of experimentally testing LHV theory as an independent hypothesis separated from the quantum formalism. It has been proved that^{[23, 24]} for the standard projective measurements all pure entangled states of two qubits violate the correlation CHSH inequality. In the case of three parties, Ací n et al.^[25] considered a Bell inequality and showed that all three-partite pure entangled states violate it. For the N-qubit case, the best-known inequality is the Mermin– Ardehali– Belinskiĭ – Klyshko (MABK) inequality, ^{[26– 28]} which is included in the Werner– Wolf– Ż ukowski– Brukner (WWZB) inequality^{[29, 30]} as a special case. It has been proved that quantum violation of this inequality increases with the number of particles. Several Bell-type inequalities have been established by using an algebraical method, ^{[31– 33]} and by finding out the stabilizer group of quantum states.^{[34, 35]} By determining facets of the correction polytope, Pitowsky et al.^[36] and Wu et al.^[37] obtained the new Bell inequalities.

In this paper, we discuss mainly the nonlocality for several canonical four-partite entangled states. In the Hilbert space ℂ ²)^{⊗ 4}, a general four-qubit pure state may be written as

where a_ijkl are the complex coefficients with the basis vectors | ijkl〉 , and A, B, C, and D are the locations of qubits, respectively. In nine classes of four-qubit entangled pure states, ^[12] we here consider the generic family

where the complex parameters a, b, c, and d satisfy the normalization condition | a| ² + | b| ² + | c| ² + | d| ² = 1. For the sake of simplicity throughout this paper we concern ourselves only with the cases where all a, b, c, and d are real. Obviously, the generic family contains four-qubit GHZ state , cluster state | cluster〉 = (| 0000〉 + | 0011〉 + | 1100〉 − | 1111〉 )/2, two-EPR-pair state | G₁₀₀₀〉 = (| 0000〉 + | 0011〉 + | 1100〉 + | 1111〉 )/2, and the maximally entangled state

We note that these states do not violate the previous four-qubit Bell inequalities^{[34, 35]} with the specified experimental settings. Next we introduce a Bell-type inequality that shows different degrees of violation when evaluated for different entangled states. In particular, the maximal violation is obtained with the four-qubit maximally entangled state | G_m〉 . At last, we present a scheme for preparing the state | G_m〉 with linear optics and cross-Kerr nonlinearities.

A remarkable feature of the present inequality is that there exists a family of the four-qubit entangled states where all states violate it. The family reads

where a and b satisfy ab > 0 and a² + b² = 1. Consider genuine four-tangle τ _{(4, 8)}, ^{[17, 18]} which is regarded as a measure of four-qubit entanglement and constructed by means of a four-qubit invariant of degree 8 expressed in terms of three-qubit invariants, we have

Then, for the state | G_ab00〉 , four-tangle is τ _{(4, 8)} = 4a²b². In the light of the result in Eq.  (11) we obtain a useful relationship between four-partite entanglement (four-tangle) and nonlocality for the family of the four-qubit entangled states | G_ab00〉

Figure  2 shows the quantum nonlocality versus four-tangle for states | G_ab00〉 . It has been shown that quantum violation varies with four-tangle τ _{(4, 8)} and when τ _{(4, 8)} = 1 (i.e., ) the maximal violation is obtained.

Fig.  2.

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	Fig.  2. Numerical results of quantum nonlocality versus four-tangle for the family of the four-qubit entangled states | Gab00〉 , which violate the present inequality.

We now discuss a possible experimental realization of the maximally entangled state | G_m〉 with linear optics and cross-Kerr nonlinearities.^{[38, 39]} The setup for creating the desired state is shown in Fig.  3. Initial state is fed by , where the qubits are encoded with the polarization modes | H〉 ≡ | 0〉 and | V〉 ≡ | 1〉 . In order to project the input state into expected subspaces, we here introduce an auxiliary single-photon state | H〉 _T as target photon to construct four controlled-NOT gates. Note that there exists the scheme for constructing near deterministic linear optical controlled-NOT gate with linear optics and cross-Kerr nonlinearities.^[38] In the present scheme, therefore, we suggest that these optical controlled-NOT gates can be realized by using several cross-Kerr media as well as linear optics. Then, a bit-flip operation (| H〉 ⇌ | V〉 ) occurs on the auxiliary photon when any of these four signal photons is horizontal polarization. At last, one makes a single-photon detection on the auxiliary photon in a basis {| H〉 , | V〉 }. When detector D_H clicks, we obtain the maximally entangled state | G_m〉 . While for D_V clicks, alternatively, one can also get the desired state by performing a bit-flip operation on any one of the four photons.

Fig.  3.

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	Fig.  3. The schematic diagram of preparation of the maximally entangled state | Gm〉 . Each polarizing beam splitter (PBS) is used to transmit | H〉 polarization photons and reflect | V〉 polarization photons. Four optical controlled-NOT gates are realized by means of cross-Kerr nonlinearities.

Finally, let us discuss several specific examples for the candidates of four-qubit maximally entangled state using the introduced Bell-type inequality. Under the specified experimental setting, we calculate expectation values of operator 𝓑 for four-qubit GHZ state, cluster state, W state, two-EPR-pair state,

presented by Yeo and Chua, ^[40]

introduced by Higuchi and Sudbery, ^[41] and

investigated by Brown et al.^[42] The result is that the inequality is not violated by these states except the W state, | 〈 W| 𝓑 | W〉 | = 2.5. Hence, the present inequality can act as a strong entanglement witness for the entangled state | G_m〉 .

In summary, we have introduced a four-qubit Bell-type inequality with the method of the stabilizer group of quantum states. Under the given experimental setting, the inequality is maximally violated by the four-qubit maximally entangled state | G_m〉 which does not violate previous four-qubit Bell-type inequality.^{[34, 35]} We have studied the quantum nonlocality of the generic family | G_abcd〉 and several four-qubit states using the inequality and shown that the inequality is also violated by the four-qubit W state and a family of four-qubit entangled states | G_ab00〉 . Also, we have derived a useful relationship between four-partite entanglement and nonlocality for the family of four-qubit entangled states | G_ab00〉 . Therefore, the results derived from the inequality indicate that it can act as a strong entanglement witness for the family | G_ab00〉 , especially for | G_m〉 . Experimentally, we notice that the present inequality is easier to evaluate because it contains only four terms. In addition, we have proposed a scheme of experimental realization of the state | G_m〉 with linear optics and cross-Kerr nonlinearities. It is straightforward to show in principle that the desired state can be prepared after a simple single-photon detection and post-selection. Finally, it is worth noting that the discovery of the entanglement-nonlocality relationship, as well as the results reported in Refs.  [43]– [46] will greatly facilitate the development of entanglement and their application in quantum information processing.^{[47– 52]}