Bell nonlocality is an intrinsic feature of quantum mechanics, which can be detected via the violation of Bell inequalities. It is therefore a fundamental question to certify Bell nonlocality from experimental data. Here we present an optimization framework to detect Bell-operator correlations by exploring flexible mappings between Bell inequalities and Hamiltonians corresponding to the Bell operators. We show that several Hamiltonian models can be mapped to new inequalities with improved classical bounds compared to the original one, enabling a more robust certification of Bell-operator correlations. From the other direction, we investigate the mapping from fixed Bell inequalities to Hamiltonians, aiming to maximize quantum violations while considering experimental imperfections. As a practical application, we experimentally exploit this method in an -like honeycomb-lattice model utilizing over 70 superconducting qubits. The successful application of this technique, together with the combination of the two directions to form an optimization loop, may open new avenues for developing more practical and noise-resilient certification techniques for Bell-operator correlations and enable broader experimental explorations.
Bell nonlocality is an intrinsic feature of quantum mechanics, which can be detected via the violation of Bell inequalities. It is therefore a fundamental question to certify Bell nonlocality from experimental data. Here we present an optimization framework to detect Bell-operator correlations by exploring flexible mappings between Bell inequalities and Hamiltonians corresponding to the Bell operators. We show that several Hamiltonian models can be mapped to new inequalities with improved classical bounds compared to the original one, enabling a more robust certification of Bell-operator correlations. From the other direction, we investigate the mapping from fixed Bell inequalities to Hamiltonians, aiming to maximize quantum violations while considering experimental imperfections. As a practical application, we experimentally exploit this method in an -like honeycomb-lattice model utilizing over 70 superconducting qubits. The successful application of this technique, together with the combination of the two directions to form an optimization loop, may open new avenues for developing more practical and noise-resilient certification techniques for Bell-operator correlations and enable broader experimental explorations.
Quantum nonlocality describes a stronger form of quantum correlation than that of entanglement. It refutes Einstein’s belief of local realism and is among the most distinctive and enigmatic features of quantum mechanics. It is a crucial resource for achieving quantum advantages in a variety of practical applications, ranging from cryptography and certified random number generation via self-testing to machine learning. Nevertheless, the detection of nonlocality, especially in quantum many-body systems, is notoriously challenging. Here, we report an experimental certification of genuine multipartite Bell-operator correlations, which signal nonlocality in quantum many-body systems, up to 24 qubits with a fully programmable superconducting quantum processor. In particular, we employ energy as a Bell-operator correlation witness and variationally decrease the energy of a many-body system across a hierarchy of thresholds, below which an increasing Bell-operator correlation depth can be certified from experimental data. We variationally prepare the low-energy state of a two-dimensional honeycomb model with 73 qubits and certify its Bell-operator correlations by measuring an energy that surpasses the corresponding classical bound with up to 48 standard deviations. In addition, we variationally prepare a sequence of low-energy states and certify their genuine multipartite Bell-operator correlations up to 24 qubits via energies measured efficiently by parity oscillation and multiple quantum coherence techniques. Our results establish a viable approach for preparing and certifying multipartite Bell-operator correlations, which provide not only a finer benchmark beyond entanglement for quantum devices, but also a valuable guide toward exploiting multipartite Bell correlations in a wide spectrum of practical applications.
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.
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.
Nonlocality shapes quantum correlations, revealed through the violation of Bell inequalities. The intersection of all valid Bell inequalities is the so-called local polytope. In multipartite systems, characterizing the local polytope quickly becomes an intractable task as the system size increases. Optimizing Bell inequalities to maximize the ratio between their quantum value and classical bound is key to understanding multipartite nonlocality. We propose a gradient-based method for this optimization. Numerical results indicate that local maxima of this ratio typically correspond to facet Bell inequalities of the local polytope. This enables an iterative search for tight and robust Bell inequalities. Applied to permutation-invariant scenarios, the method provides tight Bell inequalities with large quantum violations and facilitates experimental certification of Bell correlations without full knowledge of the local polytope. Moreover, analytical results of the maximum ratio are derived in the thermodynamic limit.
Bell nonlocality provides a robust scalable route to the efficient certification of quantum states. Here, we introduce a general framework for constructing Bell inequalities tailored to the Zd toric code for odd prime local dimensions. Selecting a suitable subset of stabilizer operators and mapping them to generalized measurement observables, we compute multipartite Bell expressions whose quantum maxima admit a sum-of-squares decomposition. We show that these inequalities are maximally violated by all states in the ground-state manifold of the Zd toric code, and determine their classical (local) bounds through a combination of combinatorial tiling arguments and explicit optimization. As a concrete application, we analyze the case of d=3 and demonstrate that the maximal violation self-tests the full qutrit toric-code subspace, up to local isometries and complex conjugation. This constitutes, to our knowledge, the first-ever example of self-testing a qutrit subspace. Extending these constructions, we further present schemes to enhance the ratio of classical--quantum bounds and thus improve robustness to experimental imperfections. Our results establish a pathway toward device-independent certification of highly entangled topological quantum matter and provide new tools for validating qudit states in error-correcting codes and quantum simulation platforms.
Bell nonlocality is the resource that enables device-independent quantum information processing tasks. It is revealed through the violation of so-called Bell inequalities, indicating that the observed correlations cannot be reproduced by any local hidden-variable model. While well explored in few-body settings, the question of which Bell inequalities are best suited for a given task remains quite open in the many-body scenario. One natural approach is to assign Bell inequalities to physical Hamiltonians, mapping their interaction graph to two-body, nearest-neighbor terms. Here, we investigate the effect of boundary conditions in a two-dimensional square lattice, which can induce different topologies in lattice systems. We find a relation between the induced topology and the Bell inequality's effectiveness in revealing nonlocal correlations. By using a combination of tropical algebra and tensor networks, we quantify their detection capacity for nonlocality. Our work can act as a guide to certify Bell nonlocality in many-qubit devices by choosing a suitable Hamiltonian and measuring its ground-state energy, a task that many quantum experiments are purposely built for.