Violating Bell's inequalities (BIs) allows one to certify the preparation of entangled states from minimal assumptions-in a device-independent manner.Finding BIs tailored to many-body correlations as prepared in present-day quantum computers and simulators is however a highly challenging endeavor.In this work, we focus on BIs violated by very coarse-grain features of the system: two-body correlations averaged over all permutations of the parties.For two-outcome measurements, specific BIs of this form have been theoretically and experimentally studied in the past, but it is practically impossible to explicitly test all such BIs.Data-driven methods-reconstructing a violated BI from the data themselves-have therefore been considered.Here, inspired by statistical physics, we develop a novel data-driven approach specifically tailored to such coarse-grain data.Our approach offers two main improvements over the existing literature: (1) it is directly designed for any number of outcomes and settings; (2) the obtained BIs are quadratic in the data, offering a fundamental scaling advantage for the precision required in experiments.This very flexible method, whose complexity does not scale with the system size, allows us to systematically improve over all previously known Bell inequalities robustly violated by ensembles of quantum spin 1/2; and to discover novel families of Bell inequalities, tailored to spin-squeezed states and many-body spin singlets of arbitrary spin-j ensembles.
Violating Bell's inequalities (BIs) allows one to certify the preparation of entangled states from minimal assumptions-in a device-independent manner.Finding BIs tailored to many-body correlations as prepared in present-day quantum computers and simulators is however a highly challenging endeavor.In this work, we focus on BIs violated by very coarse-grain features of the system: two-body correlations averaged over all permutations of the parties.For two-outcome measurements, specific BIs of this form have been theoretically and experimentally studied in the past, but it is practically impossible to explicitly test all such BIs.Data-driven methods-reconstructing a violated BI from the data themselves-have therefore been considered.Here, inspired by statistical physics, we develop a novel data-driven approach specifically tailored to such coarse-grain data.Our approach offers two main improvements over the existing literature: (1) it is directly designed for any number of outcomes and settings; (2) the obtained BIs are quadratic in the data, offering a fundamental scaling advantage for the precision required in experiments.This very flexible method, whose complexity does not scale with the system size, allows us to systematically improve over all previously known Bell inequalities robustly violated by ensembles of quantum spin 1/2; and to discover novel families of Bell inequalities, tailored to spin-squeezed states and many-body spin singlets of arbitrary spin-j ensembles.
Multipartite Bell nonlocality provides a device-independent probe of many-body quantum correlations, but its characterization is limited by the rapid growth of the underlying classical and quantum optimization problems. We develop a scalable method for constructing and certifying permutationally invariant Bell inequalities using only one- and two-body correlators. The construction gives families of inequalities with robust quantum violations for general m measurements as the number of parties N becomes large. To improve robustness against noise, we optimize the ratio of the quantum value to the classical bound for these families in the large-N limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-N ratios take simple rational values for finite m and converge to coth(1) as m→∞. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.
We address the question of assessing the number of particles sharing genuinely nonlocal correlations in a multipartite system. While the interest in multipartite nonlocality has grown in recent years, its existence in large quantum systems is difficult to confirm experimentally. This is mostly due to the inadequacy of standard multipartite Bell inequalities to many-body systems: Such inequalities usually rely on expectation values involving many parties, usually all, and require individual addressing of each party. In addition, known Bell inequalities for genuine nonlocality are composed of a number of expectation values that scales exponentially with the number of observers, which makes such inequalities impractical from the experimental point of view. In a recent work [Tura et al., Science 344, 1256 (2014)], it was shown that it is possible to detect nonlocality in multipartite systems using Bell inequalities with only two-body correlators. This opened the way for the detection of Bell correlations with trusted collective measurements through Bell correlation witnesses [Schmied et al., Science 352, 441 (2016)]. These witnesses were recently tested experimentally in many-body systems such as Bose-Einstein condensate or thermal ensembles, hence demonstrating the presence of Bell correlations with assumptions on the statistics. Here, we address the problem of detecting nonlocality depth, a notion that quantifies the number of particles sharing nonlocal correlation in a multipartite system. We introduce a general framework allowing us to derive Bell-like inequalities for nonlocality depth from symmetric two-body correlators. We characterize all such Bell-like inequalities for a finite number of parties and we show that they reveal Bell correlation depth $k\ensuremath{\le}6$ in arbitrarily large systems. We then show how Bell correlation depth can be estimated using quantities that are within reach in current experiments. On one hand, we use the standard multipartite Bell inequalities such the Mermin and Svetlichny ones to derive Bell correlations witnesses of any depth that involves only two collective measurements, one of which being the parity measurement. On the other hand, we show that our two-body Bell inequalities can be turned into witnesses of depth $k\ensuremath{\le}6$ that require measuring total spin components in certain directions. Interestingly, such a witness is violated by existing data from an ensemble of 480 atoms.
We expand the toolbox for studying Bell correlations in multipartite systems by introducing permutationally invariant Bell inequalities (PIBIs) involving few-body correlators. First, we present around twenty families of PIBIs with up to three- or four-body correlators, that are valid for an arbitrary number of particles. Compared to known inequalities, these show higher noise robustness, or the capability to detect Bell correlations in highly non-Gaussian spin states. We then focus on finding PIBIs that are of practical experimental implementation, in the sense that the associated operators require collective spin measurements along only a few directions. To this end, we formulate this search problem as a semidefinite program that embeds the constraints required to look for PIBIs of the desired form.
Quantum computers promise computational capabilities beyond the reach of efficient classical methods, raising a central question: how can one rigorously certify nonclassical behavior in large-scale quantum devices under minimal assumptions? Bell nonlocality, the strongest and most device-independent form of quantum correlation, provides a natural framework to address this question. This thesis develops mathematical and experimental tools to detect and characterize Bell nonlocality in many-body quantum systems, with a focus on scalable Bell-inequality constructions, and the use of Bell operators as practical benchmarks for quantum processors. We first introduce a tropical tensor network framework that recasts the optimization of classical bounds for multipartite Bell inequalities as a tensor contraction in tropical algebra, enabling the characterization of broad inequality families and an extension to the thermodynamic limit. Building on this, we develop a systematic theory of translation-invariant Bell inequalities in one dimension. We then study two-dimensional spin models on the square lattice, quantifying how open, periodic, and topologically nontrivial boundary conditions modify the strength of certifiable Bell violations. On the experimental side, we demonstrate certified multipartite Bell-operator correlations on a programmable superconducting processor, including a 73-qubit honeycomb model, by mapping Bell inequalities onto many-body Hamiltonians. We further introduce a bidirectional optimization framework that co-designs Bell inequalities and Hamiltonians for a given device, yielding stronger and more noise-resilient certification.
Bell nonlocality represents the ultimate consequence of quantum entanglement, fundamentally undermining the classical tenet that spatially separated degrees of freedom possess objective attributes independently of the act of their measurement. Despite its importance, probing Bell nonlocality in many-body systems is considered to be a formidable challenge, with a computational cost scaling exponentially with system size. Here we propose and validate an efficient variational scheme, based on the solution of inverse classical Ising problems, which in polynomial time can probe whether an arbitrary set of quantum data is compatible with a local theory; and, if not, it delivers the many-body Bell inequality most strongly violated by the quantum data. We use our approach to unveil new many-body Bell inequalities, violated by suitable measurements on paradigmatic quantum states (the low-energy states of Heisenberg antiferromagnets), paving the way to systematic Bell tests in the many-body realm.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.