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Canadian Journal of Physics

Tripartite entanglement and non-locality in three-qubit Greenberger–Horne–Zeilinger states with bit-flip noise

Jiaqiang Zhao, Lianzhen Cao, Yang Yang, Yingde Li, Huai-Xin Lu

Published Jun 4, 2018• 3 citations
Publisher

Abstract

Ardehali (Phys. Rev. A, 46, 5375 (1992). 10.1103/PhysRevA.46.5375 ) derived a Bell-type inequality for an n-particle system in a Greenberger–Horne–Zeilinger (GHZ) state. This inequality helps relax the detection efficiencies required for closing the detection loophole in Bell tests. Coffman et al. (Phys. Rev. A, 61, 052306 (2000). doi: 10.1103/PhysRevA.61.052306 ) gave a simplified form of tripartite entanglement of three-qubit pure states. We use theory and experiment to investigate both results using three-qubit GHZ states with different intensities of bit-flip noise. The purpose of our work is to relate the Ardehali inequality and the tripartite concurrence of the tripartite GHZ states in a noisy environment. Our results show that the experimental values are quite consistent with theoretical predictions for three-qubit GHZ states in a bit-flip noisy environment.

DOI
10.1139/cjp-2018-0076

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Ardehali (Phys. Rev. A, 46, 5375 (1992). 10.1103/PhysRevA.46.5375 ) derived a Bell-type inequality for an n-particle system in a Greenberger–Horne–Zeilinger (GHZ) state. This inequality helps relax the detection efficiencies required for closing the detection loophole in Bell tests. Coffman et al. (Phys. Rev. A, 61, 052306 (2000). doi: 10.1103/PhysRevA.61.052306 ) gave a simplified form of tripartite entanglement of three-qubit pure states. We use theory and experiment to investigate both results using three-qubit GHZ states with different intensities of bit-flip noise. The purpose of our work is to relate the Ardehali inequality and the tripartite concurrence of the tripartite GHZ states in a noisy environment. Our results show that the experimental values are quite consistent with theoretical predictions for three-qubit GHZ states in a bit-flip noisy environment.
Knowing the level of entanglement robustness against quantum bit loss or decoherence mechanisms is an important issue for any application of quantum information. Fidelity of states can be used to judge whether there is entanglement in multi-particle systems. It is well known that quantum channel security in QKD can be estimated by measuring the robustness of Bell-type inequality against noise. We experimentally investigate a new Bell-type inequality (NBTI) in the three-photon Greenberger-Horne-Zeilinger (GHZ) states with different levels of spin-flip noise. The results show that the fidelity and the degree of violation of the NBTI decrease monotonically with the increase of noise intensity. They also provide a method to judge whether there is entanglement in three-particle mixed states.
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.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the WWW type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard WWW states, under the local measurement operations carried out by a single party and ppp (p=2,3p=2,3p=2,3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.
In this paper, we investigate the genuine three-way nonlocality which is recognized as the strongest form of tripartite correlations. We consider theoretically and experimentally a series of suitable Bell-type inequalities a violation of which is sufficient for the detection of three-way nonlocality. For the generalized GHZ (gGHZ) states, it is demonstrated that they do violate tripartite Bell-type inequalities for any degree of tripartite entanglement even if they do not violate Svetlichny inequality. It implies that three-way entangled gGHZ can always exhibit genuine three-way nonlocality under the requirement of time-order-dependent principle. Furthermore, we have determined the maximal amount of noise admissible for the gGHZ states to still remain genuine three-way nonlocal.
Distributed entanglement across multi-node quantum networks is essential for a wide range of quantum technologies, including modular quantum computers, distributed sensing and metrology, and multi-party secure communication protocols. Such large-scale quantum networks will require photonic interconnects to generate and sustain entangled states across localized nodes. Previously, three-node distributed Greenberger-Horne-Zeilinger (GHZ) states have been generated between solid-state qubits and atomic ensembles, but not yet in the platform of individual atomic qubits, which can be replicated, detected, and individually controlled with high fidelity. Here we report the first fully-distributed GHZ state of qubits across a three-node quantum network of single atomic memories, using photonic interconnects. We achieve a bounded fidelity of 0.841(17)≤F≤0.881(17)0.841(17) \leq \mathcal{F} \leq 0.881(17)0.841(17)≤F≤0.881(17) at an entanglement generation rate of 0.095(5)/sec and measure a clear violation of Mermin's inequality while closing the detection loophole for the first time in a fully-distributed multipartite entangled state.