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Acta Physica Sinica

Bell-type inequality and tripartite nonlocality in three-qubit GHZ-class states

Jia-Qiang Zhao, Cao Lian-Zhen, Lu Huai-Xin, Wang Xiao-Qin

Published Jan 1, 2013• 4 citations
Publisher

Abstract

In this paper, we theoretically study the relation between concurrence and nonlocality depicted by Bell-type inequality violation of quantum mechanics prediction versus local realism prediction for the GHZ (Greenberger -Horne-Zeilinger) class states. Analytical expressions of concurrence, violations of the Mermin inequality and the Svetlichny inequality are obtained. Through numerical calculations, the relationship between entanglement and nonlocality of GHZ-class states is discussed. Our results show that the concurrence is consistent with the degree of nonlocality described by violations of the two Bell-type inequalities of GHZ-class states. The Bell operator and its parameters can obviously reveal the nonlocal features of quantum states.

DOI
10.7498/aps.62.120301

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•Jan 1, 2016•0 citations
In this paper, we theoretically study the relation between concurrence and nonlocality depicted by Bell-type inequality violation of quantum mechanics prediction versus local realism prediction for the GHZ (Greenberger -Horne-Zeilinger) class states. Analytical expressions of concurrence, violations of the Mermin inequality and the Svetlichny inequality are obtained. Through numerical calculations, the relationship between entanglement and nonlocality of GHZ-class states is discussed. Our results show that the concurrence is consistent with the degree of nonlocality described by violations of the two Bell-type inequalities of GHZ-class states. The Bell operator and its parameters can obviously reveal the nonlocal features of quantum states.
We analyze the relationship between tripartite entanglement and genuine tripartite nonlocality for three-qubit pure states in the Greenberger-Horne-Zeilinger class. We consider a family of states known as the generalized Greenberger-Horne-Zeilinger states and derive an analytical expression relating the three-tangle, which quantifies tripartite entanglement, to the Svetlichny inequality, which is a Bell-type inequality that is violated only when all three qubits are nonlocally correlated. We show that states with three-tangle less than 1/2 do not violate the Svetlichny inequality. On the other hand, a set of states known as the maximal slice states does violate the Svetlichny inequality, and exactly analogous to the two-qubit case, the amount of violation is directly related to the degree of tripartite entanglement. We discuss further interesting properties of the generalized Greenberger-Horne-Zeilinger and maximal slice states.
The violation of the Svetlichny's inequality (SI) [Phys. Rev. D 35, 3066 (1987)] is sufficient but not necessary for genuine tripartite nonlocal correlations. Here we quantify the relationship between tripartite entanglement and the maximum expectation value of the Svetlichny operator (which is bounded from above by the inequality) for the two inequivalent subclasses of pure three-qubit states: the Greenberger-Horne-Zeilinger (GHZ) class and the WWW class. We show that the maximum for the GHZ-class states reduces to Mermin's inequality [Phys. Rev. Lett. 65, 1838 (1990)] modulo a constant factor, and although it is a function of the three tangle and the residual concurrence, large numbers of states do not violate the inequality. We further show that by design SI is more suitable as a measure of genuine tripartite nonlocality between the three qubits in the WWW-class states, and the maximum is a certain function of the bipartite entanglement (the concurrence) of the three reduced states, and only when their sum attains a certain threshold value do they violate the inequality.
We investigate experimentally the violation of the Bell, Mermin and Svetlichny inequalities to local realism prediction. Using the maximum-likelihood technique to construct the density matrix and calculate the fidelity of the Greenberger-Horne-Zeilinger (GHZ) state. The values of the three inequalities are measured. The experimental resuls show that the three inequalities have different violation degrees to local realism prediction. So we can choose a suitable method of nonlocality to estimate the security of quantum channel.
As stated by S. Ghose et al. [Phys. Rev. Lett. 102, 250404 (2009)], there are certain relationships between tripartite entanglement and tripartite nonlocality for three-qubit Greenberger-Horne-Zeilinger (GHZ) class states. In the present work, we have experimentally demonstrated the theoretical results of Ghose et al. by using both three-photon generalized GHZ (GGHZ) states and maximal slice (MS) states with a count of $\ensuremath{\sim}$10/s. From the data, we have verified the agreement of the experimental violation of the Svetlichny inequality with the one predicted by quantum mechanics given the reconstructed density matrix. For the MS states, it is demonstrated that the amount of violation increases linearly following the increase of the degree of tripartite entanglement. In contrast, for GGHZ states, there is a minimal value of the violation when the degree of tripartite entanglement is 1/3. Both of the results are consist with the theoretical prediction.
We investigate genuine multiqubit nonlocality in families of entangled 3- and 4-qubit pure states by analyzing a Bell-type inequality that is violated only if all qubits are nonlocally correlated. We present detailed numerical studies of the relationship between entanglement and violation of the Svetlichny Bell-type inequality in an experimentally accessible set of 3-qubit pure states, and identify the special nonlocality property of the maximal slice states in the space of all 3-qubit pure states. We also analyze nonlocal correlations in 3-qubit generalized Greenberger–Horne–Zeilinger (GHZ) states and extend our analysis to the case of 4-qubit generalized GHZ states. We show that like the 3-qubit case, some 4-qubit generalized GHZ states do not violate a Bell inequality that tests for genuine 4-qubit nonlocality. Furthermore, the location of the boundary between the states that do violate the inequality and those that do not is the same for the 3- and 4-qubit generalized GHZ states.
Bell inequalities were formulated by John Bell to test the possible violation of local realistic theories by quantum mechanical systems. It was shown that entangled quantu-m states of multiple particles violate various Bell’s inequalities. This proved that quan-tum mechanics allows correlations between spatially separated systems that have no classical analogue. The main focus of this work is to investigate genuine multiqubit non-locality in families of entangled 3 and 4-qubit pure states by studying a Bell-type inequality that is violated only if all qubits are non-locally correlated. We numerically study the relationship between entanglement and violation of the Svetlichny Bell-type inequality. We analyze non-local correlations in 3-qubit generalized Greenberger-Hor-ne-Zeilinger (GHZ) states, maximal slice (MS) states, and W states. Our studies show that the correlations exhibited by three particles cannot in general be described by hid-den variable theories with at most two-particle non-locality. However, some 3-qubit entangled states do not violate the Svetlichny’s inequality. We then extend our analysis to 4-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states, maximal slice (MS) states, and W states. The results are similar to the 3-qubit case for GHZ and MS states. The range of parameters for which we see a violation is the same for the 3 and 4-qubit GHZ states. However, the 4-qubit W states do not violate Bell-type inequality, unlike the 3-qubit W states. Our results show the complex nature of multiqubit entang-lement and non-locality and provide tools for designing useful quantum communica-tion tasks.