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arXiv:2210.09065 · Journal of Mathematical Physics

Two convergent NPA-like hierarchies for the quantum bilocal scenario

Marc-Olivier Renou, Xiangling Xu, Laurens T. Ligthart

Published Jan 1, 2026• 3 citations
arXivHTMLPDF

Abstract

Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascués et al., New J. Phys. 10, 073013 (2008)], known as the Navascués–Pironio–Acín (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.

DOI
10.1063/5.0211008

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Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascués et al., New J. Phys. 10, 073013 (2008)], known as the Navascués–Pironio–Acín (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.
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It is difficult to establish an analytical criterion to identify the boundaries of quantum correlations, even for the simplest Bell scenario. Here, we briefly reviewed the plausible analytical criterion, and we found a way to confirm the extremal conditions from another direction. For that purpose, we analyzed the Navascués-Pironio-Acín (NPA) hierarchy to study the algebraic structure and found that the problem could not be simplified using the 1+AB level. However, considering the plausible criterion, the 1+AB and second levels for correlations were equal, and the extremal condition in the simplest Bell scenario was replaced by that in the 1+AB level. Thus, the correctness of the plausible criterion was verified, and the results demonstrated that the plausible criterion held, thereby explaining its simplicity. It seemed plausible, but now it becomes more certain.
The Navascués--Pironio--Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional hα=A0B0+A0B1+A1B0−A1B1+α(A0+B0)h_α=A_0B_0+A_0B_1+A_1B_0-A_1B_1+α(A_0+B_0)hα​=A0​B0​+A0​B1​+A1​B0​−A1​B1​+α(A0​+B0​), set T=1−αT=1-αT=1−α. Its quantum maximum satisfies [ωQ(1−T)−(4−2T)]/T3→4/3[ω_{\rm Q}(1-T)-(4-2T)]/T^3\to4/3[ωQ​(1−T)−(4−2T)]/T3→4/3, whereas every fixed NPA level satisfies [ωL(1−T)−ωQ(1−T)]/T3→+∞[ω_L(1-T)-ω_{\rm Q}(1-T)]/T^3\to+\infty[ωL​(1−T)−ωQ​(1−T)]/T3→+∞. Under the corresponding boundary rescaling, an explicit expectation of the positive operator ωQ(1−t2)I−Htω_{\rm Q}(1-t^2)I-H_tωQ​(1−t2)I−Ht​ converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.
Bounding the correlations predicted by quantum theory is an important challenge in quantum information science. Today's leading approach is semidefinite programming relaxations, but existing methods still cannot account for many relevant types of constraints. Here, we propose a semidefinite relaxation methodology that can incorporate a breadth of constraints needed in various quantum correlation problems, thereby generalising the seminal Navascués-Pironio-Acín hierarchy. It yields useful results at reasonable computational cost. We showcase the methodology and its features by using it to address five different quantum information problems. These are (i) entanglement witnessing from imperfect measurement devices, (ii) certifying measurements from fidelity-constrained sources, (iii) computing dimensionality in genuine multi-particle entangled states, (iv) benchmarking dimensionality for state preparation devices, and (v) finding uncertainty relations for nearly anti-commuting observables. These applications reflect both the usefulness and versatility of the methodology, as well as its potential for broader relevance in the field.
Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for all bipartite compiled Bell games via a newly formalized convergent sequential Navascués-Pironio-Acín (NPA) hierarchy: any polynomial-time prover's score is controlled by a finite-level hierarchy value, and finite-level convergence gives a negligible gap to the commuting quantum value, or to the tensor-product quantum value under flat optimality. Second, we develop the sequential NPA hierarchy as a computational tool of independent interest, establishing its convergence and a stopping criterion based on flatness, and identifying its conic dual as a sparse SOS hierarchy.