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SIAM Journal on Optimization

Certifying Optimality of Bell Inequality Violations: Noncommutative Polynomial Optimization through Semidefinite Programming and Local Optimization

Timotej Hrga, Igor Klep, Janez Povh

Published Apr 9, 2024• 1 citation
Publisher

Abstract

Abstract. Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large majority of a set of standard Bell inequalities, denoted A2–A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton–McCullough Positivstellensatz, the Gelfand–Naimark–Segal (GNS) construction with a novel use of the Artin–Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new “Newton chip”-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented. This technique is based on conditional expectations. Finally, noncommutative Gröbner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution.

DOI
10.1137/22m1473340

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Abstract. Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large majority of a set of standard Bell inequalities, denoted A2–A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton–McCullough Positivstellensatz, the Gelfand–Naimark–Segal (GNS) construction with a novel use of the Artin–Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new “Newton chip”-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented. This technique is based on conditional expectations. Finally, noncommutative Gröbner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution.
This talk focuses on optimization over state polynomials, i.e., polynomials in noncommuting variables and formal states of their products.An archimedean Positivstellensatz in the spirit of Putinar and Helton-McCullough is presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués-Pironio-Acín scheme for optimization of noncommutative polynomials.The motivation behind this theory arises from the study of correlations in quantum networks. Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite scenario are analyzed.To reduce the size of the semidefinite programs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited.This is based on the joint work with Igor Klep, Jurij Volčič, and Jie Wang.
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Abstract We consider the problem of optimizing the state average of a polynomial of non-commuting variables, over all states and operators satisfying a number of polynomial constraints, and over all Hilbert spaces where such states and operators are defined. Such non-commutative polynomial optimization (NPO) problems are routinely solved through hierarchies of semidefinite programming (SDP) relaxations. By formulating the general NPO problem in Lagrangian terms, we heuristically derive first-order optimality conditions via small variations in the problem variables. Although the derivation is not rigorous, it gives rise to two types of optimality conditions—state and operator—which are rigorously analyzed in the paper. Both types of conditions can be enforced through additional positive semidefinite constraints in the SDP hierarchies. State optimality conditions are shown to be satisfied by all NPO problems. For NPO problems with optimal solutions (such as, e.g., Archimedean ones) they allow enforcing a new type of constraints: namely, restricting the optimization over states to the set of common ground states of an arbitrary number of operators. Operator optimality conditions are the non-commutative analogs of the Karush–Kuhn–Tucker (KKT) conditions, which are known to hold in many classical optimization problems. In this regard, we prove that a weak form of operator optimality holds for all NPO problems; stronger versions require the problem constraints to satisfy some qualification criterion, just like in the classical case (e.g.: Mangasarian–Fromovitz constraint qualification). We test the power of the new optimality conditions by computing local properties of ground states of many-body spin systems and the maximum quantum violation of Bell inequalities.
The Navascués-Pironio-Acín (NPA) hierarchy provides a convergent sequence of semidefinite programming (SDP) relaxations for noncommutative polynomial optimisation, ubiquitous in quantum physics. However, its practical applicability is limited by the combinatorial growth in operator moments required at each level. Since not all moments contribute equally to bound tightness, selecting moments within a fixed computational budget is a relevant problem. We reframe moment selection as combinatorial subset selection and show it is governed by strong higher-order synergistic interactions among moments, quantified through a marginal synergy diagnostic adapted from complex systems theory. We develop and compare three optimisation methods: Parallel Tempering (PT), an RBM-based reinforcement learning policy, and Bayesian Optimisation (BO). On the I3322I_{3322}I3322​ Bell inequality benchmark, all three substantially outperform greedy approaches at costs around two orders of magnitude below brute force, with the RBM achieving the closest approach to optimal throughout the hard transition regime. We apply the framework to the 174 Bell inequalities in the (4,4,2,2)(4,4,2,2)(4,4,2,2) scenario, finding heterogeneous convergence behaviour across inequalities, and to the one-dimensional Heisenberg spin chain, demonstrating that physically motivated monomial bases are internally compressible and are not globally optimal in general. A budget-aware search over a broader pool improves certified bounds on long-range correlations by nearly two orders of magnitude. These results establish a scalable framework for moment selection in noncommutative polynomial optimisation, with broad applications across quantum physics and quantum information.
We present a general framework to exploit the symmetries present in the Navascu{é}s-Pironio-Ac{í}n semidefinite relaxations that approximate invariant noncommutative polynomial optimization problems. We put equal emphasis on the moment and sum-of-squares dual approaches, and provide a pedagogical and formal introduction to the Navascu{é}s-Pironio-Ac{í}n technique before working out the impact of symmetries present in the problem. Using our formalism, we compute analytical sum-of-square certificates for various Bell inequalities, and prove a long-standing conjecture about the exact maximal quantum violation of the CGLMP inequalities for dimension 3 and 4. We also apply our technique to the Sliwa inequalities in the Bell scenario with three parties with binary measurements settings/outcomes. Symmetry reduction is key to scale the applications of the NPA relaxation, and our formalism encompasses and generalizes the approaches found in the literature.
A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard variational approaches posit a wave-function ansatz and minimize over the possible states expressible by that ansatz, the problem can alternatively be formulated as a noncommutative polynomial optimization problem and treated through a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, these relaxations provide lower bounds on ground-state energies and both lower and upper bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that systematically leveraging the inherent structures of the system can substantially mitigate these scalability challenges and thus permits computing meaningful bounds for quantum spin systems on square lattices of size up to 16×1616\times1616×16.