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Physical Review A

Detecting nonlocal correlations in chain-shaped quantum networks via a deep-learning method

Ying Zhang, Jinchuan Hou, Kan He

Published Dec 16, 2024• 4 citations
Publisher

Abstract

As the demand for quantum networks and communication grows, identifying quantum nonlocality in these networks has become a crucial and challenging task. The nonconvex nature of the network local set makes traditional Bell-type inequalities inadequate for detection. Machine learning algorithms have emerged as a promising approach to address this issue. In this study we introduce a quantifier for nonlocality in chain-shaped quantum networks and employ deep-learning algorithms for its detection. We validate this method using the trilocal network, adopting an ensemble learning strategy called the stacking method, which combines four distinct deep neural networks to train a linear regression model. Our results show that machine learning effectively quantifies nonlocality in chain networks and exhibits strong generalization capabilities for Werner states.

DOI
10.1103/physreva.110.062609

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As the demand for quantum networks and communication grows, identifying quantum nonlocality in these networks has become a crucial and challenging task. The nonconvex nature of the network local set makes traditional Bell-type inequalities inadequate for detection. Machine learning algorithms have emerged as a promising approach to address this issue. In this study we introduce a quantifier for nonlocality in chain-shaped quantum networks and employ deep-learning algorithms for its detection. We validate this method using the trilocal network, adopting an ensemble learning strategy called the stacking method, which combines four distinct deep neural networks to train a linear regression model. Our results show that machine learning effectively quantifies nonlocality in chain networks and exhibits strong generalization capabilities for Werner states.
The ability to witness nonlocal correlations lies at the core of foundational aspects of quantum mechanics and its application in the processing of information. Commonly, this is achieved via the violation of Bell inequalities. Unfortunately, however, their systematic derivation quickly becomes unfeasible as the scenario of interest grows in complexity. To cope with that, here, we propose a machine learning approach for the detection and quantification of nonlocality. It consists of an ensemble of multilayer perceptrons blended with genetic algorithms achieving a high performance in a number of relevant Bell scenarios. As we show, not only can the machine learn to quantify nonlocality, but discover new kinds of nonlocal correlations inaccessible with other current methods as well. We also apply our framework to distinguish between classical, quantum, and even postquantum correlations. Our results offer a novel method and a proof-of-principle for the relevance of machine learning for understanding nonlocality.
Machine learning, the core of artificial intelligence and big data science, is one of today's most rapidly growing interdisciplinary fields. Recently, machine learning tools and techniques have been adopted to tackle intricate quantum many-body problems. In this Letter, we introduce machine learning techniques to the detection of quantum nonlocality in many-body systems, with a focus on the restricted-Boltzmann-machine (RBM) architecture. Using reinforcement learning, we demonstrate that RBM is capable of finding the maximum quantum violations of multipartite Bell inequalities with given measurement settings. Our results build a novel bridge between computer-science-based machine learning and quantum many-body nonlocality, which will benefit future studies in both areas.
Machine learning, the core of artificial intelligence and big data science, is one of today's most rapidly growing interdisciplinary fields. Recently, machine learning tools and techniques have been adopted to tackle intricate quantum many-body problems. In this Letter, we introduce machine learning techniques to the detection of quantum nonlocality in many-body systems, with a focus on the restricted-Boltzmann-machine (RBM) architecture. Using reinforcement learning, we demonstrate that RBM is capable of finding the maximum quantum violations of multipartite Bell inequalities with given measurement settings. Our results build a novel bridge between computer-science-based machine learning and quantum many-body nonlocality, which will benefit future studies in both areas.
Determining whether an observed distribution of events generated in a quantum network is Bell local, i.e., if it admits an alternative realization in terms of independent local variables, is extremely challenging. Building upon arXiv:1907.10552, we develop a software solution that parameterizes local models in networks via neural networks. This allows one to leverage optimization tools available from the machine learning community in the search of network Bell nonlocality. Our solution applies to arbitrary networks, is easy to use, and includes technical improvements that significantly increase performance compared to previous implementations. We apply it to investigate nonlocality in several networks hitherto unexplored, providing insights on the corresponding quantum nonlocal sets and suggesting concrete, promising realizations of quantum nonlocal correlations.
We address the characterization of genuine network nonlocal correlations, which remain highly challenging due to the non-convex nature of local correlations even in the distinct triangle scenario with three sources and three observers implementing one four-outcome measurement. We introduce a scalable causally inferred Bayesian learning framework called the Layered Local Hidden Variable Neural Network (Layered LHV-Net) to learn the local statistics in network Bell tests. Using this framework, we identify a new class of measurement settings that exhibit the most robust nonlocality compared to previously known measurements. Remarkably, our study shows that the nonlocality measure becomes non-zero only when the visibility of the shared Bell state exceeds 0.94, surpassing previously reported noise robustness thresholds. Further, we examine correlations where shared states originate from dissimilar sources, finding that nonlocality is observed only if all the involved states are sufficiently entangled. Finally, we analyze a scenario in which the sources are allowed to share classical randomness. We find that nonlocal correlations persist even when the sources share up to 3 units of randomness, whereas a local model reproducing the quantum correlations only becomes possible when 4 units of shared randomness are available. Apart from the results, the work succeeds in showing that quantum-informed machine learning approaches as foundational frameworks can greatly benefit the field of quantum information.
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In the above-mentioned reference, the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios has yet to be provided. The present work characterizes correlations arising due to fixed local measurements in a triangle network under a source independence assumption. Precisely speaking, a set of criteria is framed in the form of Bell-type inequalities, each of which is necessarily satisfied by trilocal correlations. Possible quantum violation of at least one criterion from the set is analyzed, which in turn points out the utility of the set of criteria to detect nonlocality (nontrilocality) in quantum triangle networks. Interestingly, measurement on a local product state basis turns out to be sufficient to generate nontrilocal correlations in some quantum networks. Noise tolerance of the detection criteria is discussed followed by a generalization of the framework for demonstrating correlations in any nnn-sided polygon where nnn is finite.