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arXiv:2502.10746 · Entropy

NPA Hierarchy and Extremal Criterion in the Simplest Bell Scenario

Satoshi Ishizaka

Published Feb 9, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3390/e27020182

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Quantum mechanics may revolutionize many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing, and metrology. Quantum cryptography, for instance, allows us to implement protocols which are device independent, i.e., they can be proven secure under fewer assumptions. These protocols rely on using devices producing nonlocal statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the Clauser-Horne-Shimony-Holt (CHSH) scenario (the simplest nontrivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two families of analytical quantum extremal points by providing solutions to two families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalization has been suggested in a sequence of works by Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details, and discuss their suitability. Based on the understanding acquired, we propose a set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on these families of quantum extremal points.

Physical Review A•Jul 24, 2023•7 citations
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.
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.
Bell nonlocality places the long-standing philosophical disputes over the nature of quantum theory on a mathematically rigorous, experimentally testable footing, positioning quantum mechanics between local realism and the most general no-signaling theories. Yet the precise quantum boundary has remained elusive even in the minimal Bell scenario, where the extreme points generating the set are known. Here we characterize this boundary through a computable necessary and sufficient criterion: a correlation is quantum if and only if it can be decomposed into an extreme quantum correlation and a classical correlation. The criterion follows from our proof that every Bell functional exhibiting quantum advantage admits a unique quantum maximizer. We further classify the finitely many maximal-face structures containing each nonlocal exposed point according to their local deterministic vertices. By sharply distinguishing physically realizable correlations from nonphysical ones, these results resolve a long-standing open problem and provide a new foundation for identifying the principles underlying quantum theory and determining the ultimate information-theoretic limits of quantum correlations.
Quantum mechanics may revolutionise many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing and metrology. Quantum cryptography for instance allows us to implement protocols which are device-independent, i.e. they can be proven security under fewer assumptions. These protocols rely on using devices producing non-local statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the CHSH scenario (the simplest non-trivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two new families of analytical quantum extremal points by providing solutions to two new families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalisation has been suggested in a sequence of works by Satoshi Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details and discuss their suitability. Based on the understanding acquired, we propose a new set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on the new families of quantum extremal points.
Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family Bs=(1−s/2)(⟨A0⟩+⟨B0⟩)+CHSHB_s=(1-s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}Bs​=(1−s/2)(⟨A0​⟩+⟨B0​⟩)+CHSH we prove: for every NPA level k≥2k\ge 2k≥2 there are an explicit rational gk>0g_k>0gk​>0 and an sk∗>0s^*_k>0sk∗​>0 with ck(s)≥4−s+gks2c_k(s)\ge 4-s+g_k s^2ck​(s)≥4−s+gk​s2 on (0,sk∗](0,s^*_k](0,sk∗​]; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally a2>1/39a_2>1/39a2​>1/39, a3>1/188a_3>1/188a3​>1/188, a4>1/641a_4>1/641a4​>1/641. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness -- a closed-form class function y∗y^*y∗ with NkTΓ(y∗)Nk=ukukTN_k^TΓ(y^*)N_k=u_k u_k^TNkT​Γ(y∗)Nk​=uk​ukT​ at every level. The mechanism forces the sign: for k≥3k\ge 3k≥3 no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.
Quantum mechanics may revolutionize many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing, and metrology. Quantum cryptography, for instance, allows us to implement protocols which are device independent, i.e., they can be proven secure under fewer assumptions. These protocols rely on using devices producing nonlocal statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the Clauser-Horne-Shimony-Holt (CHSH) scenario (the simplest nontrivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two families of analytical quantum extremal points by providing solutions to two families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalization has been suggested in a sequence of works by Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details, and discuss their suitability. Based on the understanding acquired, we propose a set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on these families of quantum extremal points.