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arXiv:1907.10552 · npj Quantum Information

A neural network oracle for quantum nonlocality problems in networks

Tamás Kriváchy, Yu Cai, Daniel Cavalcanti, Arash Tavakoli, Nicolas Gisin, Nicolas Brunner

Published Aug 21, 2020• 62 citations
arXivHTMLPDF

Abstract

Abstract Characterizing quantum nonlocality in networks is a challenging, but important problem. Using quantum sources one can achieve distributions which are unattainable classically. A key point in investigations is to decide whether an observed probability distribution can be reproduced using only classical resources. This causal inference task is challenging even for simple networks, both analytically and using standard numerical techniques. We propose to use neural networks as numerical tools to overcome these challenges, by learning the classical strategies required to reproduce a distribution. As such, a neural network acts as an oracle for an observed behavior, demonstrating that it is classical if it can be learned. We apply our method to several examples in the triangle configuration. After demonstrating that the method is consistent with previously known results, we give solid evidence that a quantum distribution recently proposed by Gisin is indeed nonlocal as conjectured. Finally we examine the genuinely nonlocal distribution recently presented by Renou et al., and, guided by the findings of the neural network, conjecture nonlocality in a new range of parameters in these distributions. The method allows us to get an estimate on the noise robustness of all examined distributions.

DOI
10.1038/s41534-020-00305-x

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Abstract Characterizing quantum nonlocality in networks is a challenging, but important problem. Using quantum sources one can achieve distributions which are unattainable classically. A key point in investigations is to decide whether an observed probability distribution can be reproduced using only classical resources. This causal inference task is challenging even for simple networks, both analytically and using standard numerical techniques. We propose to use neural networks as numerical tools to overcome these challenges, by learning the classical strategies required to reproduce a distribution. As such, a neural network acts as an oracle for an observed behavior, demonstrating that it is classical if it can be learned. We apply our method to several examples in the triangle configuration. After demonstrating that the method is consistent with previously known results, we give solid evidence that a quantum distribution recently proposed by Gisin is indeed nonlocal as conjectured. Finally we examine the genuinely nonlocal distribution recently presented by Renou et al., and, guided by the findings of the neural network, conjecture nonlocality in a new range of parameters in these distributions. The method allows us to get an estimate on the noise robustness of all examined distributions.
In the last decade, it was understood that quantum networks involving several independent sources of entanglement which are distributed and measured by several parties allowed for completely novel forms of nonclassical quantum correlations, when entangled measurements are performed. Here, we experimentally obtain quantum correlations in a triangle network structure and provide solid evidence of its nonlocality. Specifically, we first obtain the Elegant distribution proposed in [Gisin et al., Entanglement 25 years after quantum teleportation: Testing joint measurements in quantum networks, Entropy 21, 325 (2019)ENTRFG1099-430010.3390/e21030325] by performing a six-photon experiment. Then, we justify its nonlocality based on machine learning tools to estimate the distance of the experimentally obtained correlation to the local set.
Characterizing the set of distributions that can be realized in the triangle network is a notoriously difficult problem. In this work, we investigate inner approximations of the set of local (classical) distributions of the triangle network. A quantum distribution that appears to be nonlocal is the elegant joint measurement (EJM) [], which motivates us to study distributions having the same symmetries as the EJM. We compare analytical and neural-network-based inner approximations and find a remarkable agreement between the two methods. Using neural network tools, we also conjecture network Bell inequalities that give a trade-off between the levels of correlation and symmetry that a local distribution may feature. Our results considerably strengthen the conjecture that the EJM is nonlocal.
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.
In recent years, the study of Bell nonlocality has been generalized to quantum networks, where multiple independent sources distribute physical systems to distant parties who perform local measurements. In this context, a central open question is to identify the minimal network configuration in which quantum resources produce Bell nonlocal correlations. Here we address this question and show that quantum nonlocality is possible in the triangle network where the parties have no input choices and produce only binary-valued outcomes. To do so, we start by identifying a family of target distributions and proving their nonlocality. Next, we construct an explicit quantum model that reproduces the target distributions to machine precision. For this, we develop an efficient method for parameterizing quantum distributions in networks, inspired by the formalism of higher-order quantum operations. When considering the number of observed variables and their cardinality, this constitutes the smallest scenario possible that supports quantum nonlocality in networks. Moreover, by analyzing the explicit quantum model, we obtain new insights into how nonlocal distributions can be generated in quantum networks.
The study of nonlocality in scenarios that depart from the bipartite Einstein-Podolsky-Rosen setup is allowing one to uncover many fundamental features of quantum mechanics. Recently, an approach to building network-local models based on machine learning led to the conjecture that the family of quantum triangle distributions of [Renou et al., Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] did not admit triangle-local models in a larger range than the original proof. We prove part of this conjecture in the affirmative. Our approach consists of reducing the family of original, four-outcome distributions to families of binary-outcome ones, and then using the inflation technique to prove that these families of binary-outcome distributions do not admit triangle-local models. This constitutes the first successful use of inflation in a proof of quantum nonlocality in networks for distributions whose nonlocality could not be proved with alternative methods. Moreover, we provide a method to extend proofs of network nonlocality in concrete distributions of a parametrized family to continuous ranges of the parameter. In the process, we produce a large collection of network Bell inequalities for the triangle scenario with binary outcomes, which are of independent interest.
Quantum networks allow for novel forms of quantum nonlocality. By exploiting the combination of entangled states and entangled measurements, strong nonlocal correlations can be generated across the entire network. So far, all proofs of this effect are essentially restricted to the idealized case of pure entangled states and projective local measurements. Here we present noise-robust proofs of network quantum nonlocality, for a class of quantum distributions on the triangle network that are based on entangled states and entangled measurements. The key ingredient is a result of approximate rigidity for local distributions that satisfy the so-called “parity token counting'' property with high probability. Our methods can be applied to any type of noise. As illustrative examples, we consider quantum distributions obtained with imperfect sources and obtain a noise robustness up to ∼80% for dephasing noise and up to ∼0.5% for white noise. Additionally, we prove that all distributions in the vicinity of some ideal quantum distribution are nonlocal, with a bound on the total-variation distance ∼0.25% . Our work opens interesting perspectives towards the practical implementation of quantum network nonlocality.