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arXiv:2606.19033

Contextuality as a Diagnostic of Translation-Symmetry Breaking in Translation-Invariant 1D Hamiltonians

Xiao Zeng, Kaiyan Yang, Lingxia Zhang, Zizhu Wang

quant-phSubmitted Jun 17, 2026• Announced Jun 18, 2026• 0 citations
v1 · Jun 17, 2026v2 · Aug 11, 2026
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Abstract

Bell- and contextuality-type inequalities have become practical probes of many-body quantum correlations, often involving only few-body correlators and quantities with a direct Hamiltonian interpretation such as an energy density. Here we show that, in infinite one-dimensional translation-invariant chains, contextuality can acquire a genuinely thermodynamic meaning: within the witness families studied, the maximal quantum violation coincides with spontaneous breaking of one-site translation symmetry, producing strictly $p$-periodic ground states with $p>1$. Along natural continuous interpolations between classical-bound and quantum-optimal Hamiltonians, the classical bound marks a symmetry-breaking point where competing classical periodicities are lifted in favor of a unique quantum-selected period. At the quantum optimum, the studied families admit exact finite-size reductions: a translation-invariant contextuality witness induces a $p$-site periodic-boundary-condition inequality with identical classical and quantum bounds (hence no loss under reduction), and in several cases the resulting finite inequalities are tight. This reduction turns an infinite-chain contextuality certification into a compact, hardware-testable benchmark on a small ring, requiring only local energy measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using a translation-invariant adaptation of semidefinite-program hierarchies together with variational matrix-product-state algorithms.

Comments
20 pages, 6 figures

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Bell- and contextuality-type inequalities have become practical probes of many-body quantum correlations, often involving only few-body correlators and quantities with a direct Hamiltonian interpretation such as an energy density. Here we show that, in infinite one-dimensional translation-invariant chains, contextuality can acquire a genuinely thermodynamic meaning: within the witness families studied, the maximal quantum violation coincides with spontaneous breaking of one-site translation symmetry, producing strictly ppp-periodic ground states with p>1p>1p>1. Along natural continuous interpolations between classical-bound and quantum-optimal Hamiltonians, the classical bound marks a symmetry-breaking point where competing classical periodicities are lifted in favor of a unique quantum-selected period. At the quantum optimum, the studied families admit exact finite-size reductions: a translation-invariant contextuality witness induces a ppp-site periodic-boundary-condition inequality with identical classical and quantum bounds (hence no loss under reduction), and in several cases the resulting finite inequalities are tight. This reduction turns an infinite-chain contextuality certification into a compact, hardware-testable benchmark on a small ring, requiring only local energy measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using a translation-invariant adaptation of semidefinite-program hierarchies together with variational matrix-product-state algorithms.
We study local marginals in one-dimensional translation-invariant quantum systems that may hide finite-period structure. Given an nnn-site reduced density matrix, we ask whether it can be obtained by repeating a finite ppp-site block state along the chain and averaging over the ppp lattice translations. This defines a symmetrized block-product periodic marginal problem, which provides a route both to diagnosing hidden periodic order from local data and to upper bounding ground-state energy densities of infinite translation-invariant local Hamiltonians. We develop two complementary methods. The first is a semidefinite-programming relaxation based on block permutation symmetry and positive partial transpose constraints, which outer-approximates the convex hull of such marginals and yields certified infeasibility tests. The second is a symmetrized matrix product state ansatz, which constructs explicit block-product periodic states and gives variational upper bounds. We benchmark the framework on the Majumdar-Ghosh model, transverse-field Ising, XX, XXZ, and contextuality-related spin models. The results show that the method captures the expected finite-period structure in exactly solvable cases and gives systematically improving variational energies as the period and bond dimension increase. We also formulate a periodic-NPA relaxation for translation-invariant contextuality witnesses and recover the known quantum limits in the tested examples.
Contextuality, one of the strongest forms of quantum correlations, delineates the boundary between the quantum world and the classical one. Recent advances show that some translation-invariant contextuality witnesses are maximally violated by ground states and local observables of infinite one-dimensional translation-invariant Hamiltonians. However, these models all have local Hilbert space dimension larger than two, making the study of their ground states behavior difficult on current qubit-based platforms. In this work, we focus on the cost of simulating their 3-site reduced density matrices using qubit-based parameterized quantum circuits. The local approximations are purified then encoded into permutation-symmetric qubit states. By developing a universal set of permutation-symmetry preserving qubit-based gates, we assess the accuracy of simulating the purified local ground states against fixed classical and quantum resources. Results reveal that more contextual ground states with lower energy density are easier to simulate under identical resources. The authors encode qutrit contextual-Hamiltonian ground states into permutation-symmetric qubits with symmetry-preserving gates and use parameterized quantum circuits to variationally simulate their purified 3-site reduced states. They find that stronger contextuality reduces simulation cost, although all contextual states cost more than non-contextual ones.
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 mmm measurements as the number of parties NNN 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-NNN limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-NNN ratios take simple rational values for finite mmm and converge to coth⁡(1)\coth(1)coth(1) as m→∞m\to\inftym→∞. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.
Abstract In recent years there has been a growing interest in treating many-body systems as Bell scenarios, where lattice sites play the role of distant parties and only near-neighbor statistics are accessible. We investigate contextuality arising from three Bell scenarios in infinite, translation-invariant 1D models: nearest-neighbor with two dichotomic observables per site; nearest- and next-to-nearest neighbor with two dichotomic observables per site, and nearest-neighbor with three dichotomic observables per site. For the first scenario, we give strong evidence that it cannot exhibit contextuality, not even in non-signaling physical theories beyond quantum mechanics. For the second one, we identify several low-dimensional models that reach the ultimate quantum limits, paving the way for self-testing ground states of quantum many-body systems. For the last scenario, which generalizes the Heisenberg model, we give strong evidence that, in order to exhibit contextuality, the dimension of the local quantum system must be at least 3.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three and four-partite Bell inequalities constructed from one and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three- and four-partite Bell inequalities constructed from one- and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.