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arXiv:quant-ph/0405119 · Physical Review A

Nonlocality of cluster states of qubits

Valerio Scarani, Antonio Acín, Emmanuel Schenck, Markus Aspelmeyer

Published Apr 18, 2005• 195 citations
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Abstract

We investigate cluster states of qubits with respect to their nonlocal properties. We demonstrate that a Greenberger-Horne-Zeilinger (GHZ) argument holds for any cluster state: more precisely, it holds for any partial, thence mixed, state of a small number of connected qubits (five, in the case of one-dimensional lattices). In addition, we derive a Bell inequality that is maximally violated by the four-qubit cluster state and is not violated by the four-qubit GHZ state.

DOI
10.1103/physreva.71.042325

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We investigate cluster states of qubits with respect to their nonlocal properties. We demonstrate that a Greenberger-Horne-Zeilinger (GHZ) argument holds for any cluster state: more precisely, it holds for any partial, thence mixed, state of a small number of connected qubits (five, in the case of one-dimensional lattices). In addition, we derive a Bell inequality that is maximally violated by the four-qubit cluster state and is not violated by the four-qubit GHZ state.
We derive a Bell inequality for testing violation of local realism. Quantum nonlocality of several four-qubit states is investigated. These include the Greenberger-Zeilinger-Horne (GHZ) state, WWW state, linear cluster state, and the state $\ensuremath{\mid}\ensuremath{\chi}⟩$ that has recently been proposed in [Phys. Rev. Lett. 96, 060502 (2006)]. The Bell inequality is optimally violated by $\ensuremath{\mid}\ensuremath{\chi}⟩$ but not violated by the GHZ state. The linear cluster state also violates the Bell inequality though not optimally. The state $\ensuremath{\mid}\ensuremath{\chi}⟩$ can thus be discriminated from the linear cluster state by using the inequality. Different aspects of four-partite entanglement are also studied by considering the usefulness of a family of four-qubit mixed states as resources for two-qubit teleportation. Our results generalize those in [Phys. Rev. Lett. 72, 797 (1994)].
In this paper, we show that there are eight distinct forms of the Greenberger-Horne-Zeilinger (GHZ) argument for the four-qubit cluster state ∣ϕ4>|ϕ_4>∣ϕ4​> and forty eight distinct forms for the five-qubit cluster state ∣ϕ5>|ϕ_5>∣ϕ5​> in the case of the one-dimensional lattice. The proof is obtained by regarding the pair qubits as a single object and constructing the new Pauli-like operators. The method can be easily extended to the case of the N-qubit system and the associated Bell inequalities are also discussed. Consequently, we present a complete construction of the GHZ theorem for the cluster states of N-qubit in the case of the one-dimensional lattice.
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.
Cluster states are multiparticle entangled states with special entanglement properties particularly suitable for quantum computation. It has been shown that cluster states can exhibit Greenberger-Horne-Zeilinger (GHZ) -type nonlocality even when some of their qubits have been lost. In the present work, we generated a four-photon mixed state, which is equivalent to the partial, qubit-loss state of an NNN-qubit cluster state up to some local transformations. By using this mixed state, we then realize a GHZ-type violation of local realism. Our results not only demonstrate a mixed state's GHZ-type nonlocality but also exhibit the robustness of cluster states under qubit-loss conditions.
Mixed states appear naturally in experiment over pure states. So for studying different notions of nonlocality and their relation with entanglement in realistic scenarios, one needs to consider mixed states. In a recent paper [Phys. Rev. Lett. 108, 020502 (2012)], a complete characterization of entanglement of an entire class of mixed three-qubit states with the same symmetry as the Greenberger-Horne-Zeilinger state, known as GHZ-symmetric states, has been achieved. In this paper we investigate different notions of nonlocality of the same class of states. By finding the analytical expressions of maximum violation value of most efficient Bell inequalities we obtain the conditions of standard nonlocality and genuine nonlocality of this class of states. Also the relation between entanglement and nonlocality is discussed for this class of states. Interestingly, genuine entanglement of GHZ-symmetric states is necessary to reveal standard nonlocality. However, it is not sufficient to exploit the same.