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arXiv:quant-ph/0311105 · Physical Review A

Relation between entanglement measures and Bell inequalities for three qubits

Clive Emary, C. W. J. Beenakker

Published Mar 23, 2004• 37 citations
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Abstract

For two qubits in a pure state there exists a one-to-one relation between the entanglement measure (the concurrence $\mathcal{C}$) and the maximal violation $\mathcal{M}$ of a Bell inequality. No such relation exists for the three-qubit analog of $\mathcal{C}$ (the tangle $\ensuremath{\tau}$), but we have found that numerical data is consistent with a simple set of upper and lower bounds for $\ensuremath{\tau}$ given $\mathcal{M}$. The bounds on $\ensuremath{\tau}$ become tighter with increasing $\mathcal{M}$, so they are of practical use. The Svetlichny form of the Bell inequality gives tighter bounds than the Mermin form. We show that the bounds can be tightened further if the tangle is replaced by an entanglement monotone that can identify both the W state and the Greenberger-Horne-Zeilinger state.

DOI
10.1103/physreva.69.032317

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For two qubits in a pure state there exists a one-to-one relation between the entanglement measure (the concurrence C\mathcal{C}C) and the maximal violation M\mathcal{M}M of a Bell inequality. No such relation exists for the three-qubit analog of C\mathcal{C}C (the tangle \ensuremathτ\ensuremath{\tau}\ensuremathτ), but we have found that numerical data is consistent with a simple set of upper and lower bounds for \ensuremathτ\ensuremath{\tau}\ensuremathτ given M\mathcal{M}M. The bounds on \ensuremathτ\ensuremath{\tau}\ensuremathτ become tighter with increasing M\mathcal{M}M, so they are of practical use. The Svetlichny form of the Bell inequality gives tighter bounds than the Mermin form. We show that the bounds can be tightened further if the tangle is replaced by an entanglement monotone that can identify both the W state and the Greenberger-Horne-Zeilinger state.
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 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.
Entangled states of multiple qubits can violate Bell-type inequalities indicating nonlocal behavior of multiqubit quantum correlations. We analyze the relation between multipartite entanglement and genuine multipartite nonlocality, characterized by Svetlichny inequality violations, for two families of N − qubit states. We show that for the generalized Greenberger-Horne-Zeilinger family of states, Svetlichny inequality is not violated when the n − tangle is less than 1 / 2 for any even number of qubits. On the other hand, the maximal slice states always violate the Svetlichny inequality when n − tangle is nonzero, and the violation increases monotonically with tangle. Our work generalizes the relations between tangle and Svetlichny inequality violations previously derived for three qubits.
We propose an alternative evaluation of quantum entanglement by measuring the maximum violation of the Bell’s inequality without information of the reduced density matrix of a system. This proposal is demonstrated by bridging the maximum violation of the Bell’s inequality and a concurrence of a pure state in an n-qubit system, in which one subsystem only contains one qubit and the state is a linear combination of two product states. We apply this relation to the ground states of four qubits in the Wen-Plaquette model and show that they are maximally entangled. A topological entanglement entropy of the Wen-Plaquette model could be obtained by relating the upper bound of the maximum violation of the Bell’s inequality to the generalized concurrence of a pure state with respect to different bipartitions.
There are different families of inequalities that can be used to characterize the entanglement of multiqubit entangled states by the violation of quantum mechanics prediction versus local realism prediction. In a noisy environment, the violation of different inequalities distinguishes a direct from a noise-free environment. That is, each inequality has a different robustness against noise. We investigate theoretically and experimentally this proposition with the Mermin inequality, Bell inequality, and Svetlichny inequality using three-qubit GHZ states for different levels of noise. Our purpose is to determine which one of the inequalities is more robust against noise and thus more suitable to characterize entanglement of states. Our results show that the Mermin inequality is the most robust against stronger noise and is, thus, more suitable for characterizing the entanglement of three-qubit GHZ states in a noisy environment.
It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.