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.
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.
When quantum systems are shared by multiple parties in a network, the measurement outcomes of the parties can exhibit non-classical correlations, i.e., correlations that cannot be obtained if the parties shared classical systems instead. This phenomenon is known as quantum nonlocality and is typically demonstrated in the Bell scenario. However, the Bell scenario is fundamentally simpler to investigate than general networks, since the latter come with non-convex optimization problems that are often intractable. The triangle network is one of the simplest networks exhibiting this non-convexity due to the presence of three independent sources. Although some special cases of quantum nonlocality are known in the triangle network, general methods to certify classical incompatibility are still lacking, which suggests that our understanding of networks is still rather limited. For instance, the Elegant Joint Measurement (EJM) distribution is a simple and highly-symmetric outcome distribution that can be obtained with quantum systems and measurements in the triangle network. This distribution was conjectured to be non-classical eight years ago. In this article, we provide the first proof of non-classicality of the EJM distribution. To do so, we show how to combine inflation, a causal inference technique, with powerful symmetry reductions and Frank-Wolfe algorithms for large-scale optimization. We then use these methods to obtain computer-assisted proofs of non-classicality in exact arithmetic.
Nonlocal correlations created in networks with multiple independent sources enable surprising phenomena in quantum information and quantum foundations. The presence of independent sources, however, makes the analysis of network nonlocality challenging, and even in the simplest nontrivial scenarios a complete characterization is lacking. In this work we study one of the simplest of these scenarios, namely that of distributions invariant under permutations of parties in the minimal triangle network, which features no inputs and binary outcomes. We perform an exhaustive search for triangle-local models, and from it we infer analytic expressions for the boundaries of the set of distributions that admit such models, which we conjecture to be all the tight Bell inequalities for the scenario. Armed with them and with improved outer approximations of the set, we provide insights on the existence of a classical-quantum gap in the triangle network with binary outcomes.
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.
Defining nonlocality in a no-input closed quantum network scenario is a new area of interest nowadays. Gisin, in[Entropy 21, 325 (2019)], proposed a possible condition for non-tri-locality of the trivial no-input closed network scenario, triangle network, by introducing a new kind of joint measurement bases and a probability bound. In[npj Quantum Information (2020) 6:70] they found a shred of numerical evidence in support of Gisin's probability bound. Now based on that probability bound, we find the nature of the correlation in a triangle network scenario. We here observe how far the probability lies from that Gisin's bound with every possible combination of entangled and local pure states distributed from three independent quantum sources. Here we use the generalized Elegant Joint Measurements bases for each party and find that there is a dependency of non-locality on the entanglement of these joint measurement bases. We also check the probability bound for the polygon structure.
Network nonlocality allows one to demonstrate nonclassicality in networks with fixed joint measurements, that is, without random measurement settings. The simplest network in a loop, the triangle, with four outputs per party is especially intriguing. The ``elegant distribution'' [Gisin, Entropy 21, 325 (2019)] still resists analytic proofs, despite its many symmetries. In particular, this distribution is invariant under any output permutation. The Finner inequality, which holds for all local and quantum distributions, has been conjectured to be also valid for all no-signaling distributions with independent sources (NSI distributions). Here we provide evidence that this conjecture is false by constructing a four-output network box that violates the Finner inequality and prove that it satisfies all NSI inflations up to the enneagon. As a first step toward the proof of the nonlocality of the elegant distribution, we prove the nonlocality of the distributions that saturates the Finner inequality by using geometrical arguments.
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.