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AIP conference proceedings

Analysis of a tripartite Bell inequality for 3-qubit states

Shohini Ghose, Neil E. Sinclair, Shantanu Debnath, A. Kabra, Alexander I. Lvovsky

Published Jan 1, 2009• 0 citations
Publisher

Abstract

We consider 3‐qubit entangled pure states of the form α|000〉+β|111〉 and explore nonlocal correlations between the qubits by analyzing a Bell inequality belonging to the Mermin‐Klyshko (MK) set of N‐qubit inequalities. We find an expression for the maximum expectation value of the MK Bell operator and show that it does not increase smoothly with the amount of entanglement between the qubits. A subset of these tripartite entangled states does not violate the bound of 2 imposed by local hidden variable theories.

DOI
10.1063/1.3131389

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We consider 3‐qubit entangled pure states of the form α|000〉+β|111〉 and explore nonlocal correlations between the qubits by analyzing a Bell inequality belonging to the Mermin‐Klyshko (MK) set of N‐qubit inequalities. We find an expression for the maximum expectation value of the MK Bell operator and show that it does not increase smoothly with the amount of entanglement between the qubits. A subset of these tripartite entangled states does not violate the bound of 2 imposed by local hidden variable theories.
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.
Abstract We provide an analytical tripartite-study from the generalized R-matrix. It provides the upper bound of the maximum violation of Mermin’s inequality. For a generic 2-qubit pure state, the concurrence or R-matrix characterizes the maximum violation of Bell’s inequality. Therefore, people expect that the maximum violation should be proper to quantify Quantum Entanglement. The R-matrix gives the maximum violation of Bell’s inequality. For a general 3-qubit state, we have five invariant entanglement quantities up to local unitary transformations. We show that the five invariant quantities describe the correlation in the generalized R-matrix. The violation of Mermin’s inequality is not a proper diagnosis due to the non-monotonic behavior. We then classify 3-qubit quantum states. Each classification quantifies Quantum Entanglement by the total concurrence. In the end, we relate the experiment correlators to Quantum Entanglement.
It is presented a criterion of efficiently detecting all pure entangled states of n qubits by observing fluctuation of Bell's operators in place of expectation in Bell's inequalities. The entanglement witnesses are some of MK Bell operators corresponding to Mermin-Klyshko inequalities, which are easily constructed by resolving a maximum problem and using a computable method, respectively. We illustrate generalized GHZ states, generalized W and pure three-qubit states, respectively, for seeing how good our criterion is to detect the entanglement of nnn qubits. Therefore, we eventually resolve the problem of efficiently detecting pure entangled states of n qubits.
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.
Mermin's inequality is the generalization of the Bell-CHSH inequality for three qubit states. The violation of the Mermin inequality guarantees the fact that there exists quantum non-locality either between two or three qubits in a three qubit system. In the absence of an analytical result to this effect, in order to check for the violation of Mermin's inequality one has to perform a numerical optimization procedure for even three qubit pure states. Here we derive an analytical formula for the maximum value of the expectation of the Mermin operator in terms of eigenvalues of symmetric matrices, that gives the maximal violation of the Mermin inequality for all three qubit pure and mixed states.
Quantum entanglement and non-locality are two special aspects of quantum correlations. The relationship between multipartite entanglement and non-locality is at the root of the foundations of quantum mechanics but there is still no general quantitative theory. In order to address this issue we analyze the relationship between tripartite non-locality and tripartite entanglement measure, called the three-tangle. We describe the states which give the extremal quantum values of a Bell-type inequality for a given value of the tripartite entanglement. Moreover, we show that such extremal states can be reached if one introduced an appropriate order induced by the three-π entanglement measure. Finally, we derive an analytical expression relating tripartite entanglement to the maximal violations of the Bell-type inequalities.