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Physical Review A

Svetlichny’s approach to detecting genuine multipartite entanglement in arbitrarily-high-dimensional systems by a Bell-type inequality

Dong-Ling Deng, Zi-Sui Zhou, Jing‐Ling Chen

Published Aug 13, 2009• 8 citations
Publisher

Abstract

Based on the assumption of partial separability, we derive Bell-type inequalities for $N$ $d$-dimensional (qudit) systems, which can detect experimentally genuine multipartite entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. We also introduce the concept ``resolution'' to measure the strength of these inequalities to distinguish full entanglement from partial entanglement when white noise is added to the system.

DOI
10.1103/physreva.80.022109

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Based on the assumption of partial separability, we derive Bell-type inequalities for NNN ddd-dimensional (qudit) systems, which can detect experimentally genuine multipartite entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. We also introduce the concept ``resolution'' to measure the strength of these inequalities to distinguish full entanglement from partial entanglement when white noise is added to the system.
We derive a set of Bell-type inequalities for arbitrarily-high-dimensional systems based on the assumption of partial separability in the hybrid local-nonlocal hidden-variable model. Partially entangled states would not violate the inequalities, and thus, upon violation, these Bell-type inequalities are sufficient conditions to detect the full NNN-particle entanglement and invalidity of the hybrid local-nonlocal hidden-variable description.
The structure of Bell-type inequalities detecting genuine multipartite nonlocality, and hence detecting genuine multipartite entanglement, is investigated. We first present a simple and intuitive approach to Svetlichny's original inequality, which provides a clear understanding of its structure and of its violation in quantum mechanics. Based on this approach, we then derive a family of Bell-type inequalities for detecting genuine multipartite nonlocality in scenarios involving an arbitrary number of parties and systems of arbitrary dimension. Finally, we discuss the tightness and quantum mechanical violations of these inequalities.
Abstract This paper investigates the problem of nonlocality detection for high-dimensional quantum states. To overcome the complexity of measurement settings and Bell-operator optimization in high-dimensional systems, we propose a nonlocality detection framework based on dimensional reduction and chained Bell inequalities. By employing local projection operators, arbitrary high-dimensional bipartite quantum states are mapped onto effective two-qubit subsystems. We prove that if any projected subsystem violates a chained Bell inequality, then the original high-dimensional quantum state must be nonlocal. Furthermore, the framework is generalized to multipartite quantum systems. By projecting multipartite high-dimensional states onto effective multiqubit states and applying Svetlichny inequalities, sufficient criteria for detecting genuine multipartite nonlocality are established. Our approach significantly reduces the complexity of nonlocality detection in high-dimensional systems and provides a scalable theoretical framework for studying nonlocal correlations in complex quantum systems.
Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust device-independent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled Greenberger-Horne-Zeilinger states, showing that the method applies equally well to less standard states.
We introduce a general framework for detecting genuine multipartite entanglement and non-full-separability in multipartite quantum systems of arbitrary dimensions based on correlation tensors. Regarding genuine multipartite entanglement, our conditions are comparable to those of previous approaches in the case of qubits, while they show particular strength in the relatively unexplored case of higher-dimensional systems. In the case of non-full-separability, our conditions prove to be advantageous in situations where more than two-body correlations are relevant, where most previous conditions turned out to be weak. Moreover, they allow for the detection of fully bound entangled states. Finally, we also discuss experimentally friendly ways of implementing our conditions, which are based on directly measurable quantities.
In a multipartite setting, it is possible to distinguish quantum states that are genuinely nnn-way entangled from those that are separable with respect to some bipartition. Similarly, the nonlocal correlations that can arise from measurements on entangled states can be classified into those that are genuinely nnn-way nonlocal, and those that are local with respect to some bipartition. Svetlichny introduced an inequality intended as a test for genuine tripartite nonlocality. This work introduces two alternative definitions of nnn-way nonlocality, which we argue are better motivated both from the point of view of the study of nature, and from the point of view of quantum information theory. We show that these definitions are strictly weaker than Svetlichny's, and introduce a series of suitable Bell-type inequalities for the detection of three-way nonlocality. Numerical evidence suggests that all three-way entangled pure quantum states can produce three-way nonlocal correlations.