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The non-locality of cluster states

Valerio Scarani, Antonio Acín, Emmanuel Schenck, Markus Aspelmeyer

Published May 20, 2004• 0 citations
OpenAlex

Abstract

We investigate cluster states of qubits under the standpoint of their non-local properties. We demonstrate that a Greenberger-Horne-Zeilinger (GHZ) argument holds for any cluster state: more precisely, it holds for any partial, thence {\\em mixed}, state of a small number of connected qubits (five, in the case of one-dimensional lattices). For the 4-qubit cluster state, we also provide a new Bell inequality that is maximally violated by this state and is not violated by the four-qubit GHZ state.

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We investigate cluster states of qubits under the standpoint of their non-local properties. We demonstrate that a Greenberger-Horne-Zeilinger (GHZ) argument holds for any cluster state: more precisely, it holds for any partial, thence {\em mixed}, state of a small number of connected qubits (five, in the case of one-dimensional lattices). For the 4-qubit cluster state, we also provide a new Bell inequality that is maximally violated by this 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.
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.
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.
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.