A phase transition in the exactness of the NPA hierarchy at the critical doubly-tilted CHSH functional

Abstract: Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $$B_{αβ}=α\langle A_0\rangle+β\langle B_0\rangle+\mathrm{CHSH}$$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $α+β=2$. We quantify the mechanism on the symmetric slice $s=2-α-β$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s<sup>3/6-s<sup>4/36+O(s<sup>5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s<sup>2+O(s<sup>3)$, with the almost-quantum coefficient computed exactly, $a_{1+AB}=3/64$; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence $(a_k)$ - proven for every $k$ in the companion paper. We prove the supercritical side completely: for all $α,β\ge 1$ and every level the hierarchy is exact, via three explicit rational certificates realizing an affine identity. The hierarchy's exactness thus undergoes a phase transition at the critical line. On the subcritical side we certify the first four levels in exact arithmetic (rational pseudo-moments beating $c_Q$, confirmed by Sturm's theorem). We identify the exact mechanism: rescaled to the critical corner, the limiting obstruction is the Motzkin polynomial, the classical nonnegative-but-not-sum-of-squares form, so the finite-level failure sits in the restricted-certificate regime. The phase boundary has a precise geometric reading via Nie's finite-convergence theorem and Marshall's boundary Hessian condition: a self-tested optimum is finitely NPA-certifiable whenever its boundary Hessian is nondegenerate (contact order two), which holds for the single tilt and fails exactly at the doubly-tilted cubic touch. Three verified errata in the published polynomial system of Gigena et al. are documented.

• The paper rigorously characterizes a phase transition where finite-level NPA certificates fail to capture the exact quantum bound for the doubly-tilted CHSH functional.
• It provides explicit rational certificates and sharp Puiseux expansions that quantify the cubic degeneracy and nonlinear overshoot in the phase transition.
• The results clarify the limits of device-independent certification by revealing noncommutative obstacles in semidefinite relaxations and setting a framework for future studies.

Phase Transition in NPA Hierarchy Exactness for the Doubly-Tilted CHSH Functional

Introduction

The paper addresses the exactness of the Navascués-Pironio-Acín (NPA) hierarchy in bounding quantum nonlocality for the doubly-tilted CHSH functional. Gigena et al. had identified a potential phase transition where finite-level NPA certificates fail to match the exact quantum bound as the parameters approach a critical line in the functional's parameter space. This work quantifies the mechanism underlying this phase transition, characterizes its geometry, and provides explicit arithmetic certificates for both sides of the transition, thereby resolving pivotal open questions in the certification of quantum nonlocality.

Setting and Known Results

The doubly-tilted CHSH functional is given by $$B_{\alpha\beta} = \alpha\langle A_0\rangle + \beta\langle B_0\rangle + \mathrm{CHSH}$$, with local bound $c_L = 2 + \alpha + \beta$. In the quantum advantage region $0<\alpha+\beta<2$, prior results established $c_Q(\alpha, \beta)$ as the exact quantum maximum, obtainable as the largest root of an explicit polynomial system. Near the critical line $\alpha + \beta = 2$, Gigena et al. observed unbounded growth in the NPA level required for exactness, and this paper provides precise quantitative analysis and explicit certificates for both the supercritical and subcritical regions.

Quantum Expansion and Cubic Law

A Puiseux expansion of the corrected polynomial system shows the quantum maximum leaves the local bound cubically: $$c_Q(s) = 4 - s + \frac{s^3}{6} - \frac{s^4}{36} + O(s^5)$$ for $s = 2 - \alpha - \beta$ and $\alpha = \beta$. This cubic flatness at the critical point ($s=0$) reflects a degenerate self-testing realization, and is the structural cause for finite-level NPA failure; the quantum moment matrix collapses linearly to compatibility, inhibiting the power of SOS/NPA certificates at low degree.

Finite-Level NPA Overshoot: Quantitative Certificates

Each finite NPA level $k$ overshoots the quantum value quadratically: $c_L = 2 + \alpha + \beta$0, with $c_L = 2 + \alpha + \beta$1 (proven in exact arithmetic), $c_L = 2 + \alpha + \beta$2, $c_L = 2 + \alpha + \beta$3. The positivity of $c_L = 2 + \alpha + \beta$4 for any finite $c_L = 2 + \alpha + \beta$5 implies level $c_L = 2 + \alpha + \beta$6 is not exact near the critical line. Explicit certificates constructed in rational arithmetic confirm this for the first four levels, at specific $c_L = 2 + \alpha + \beta$7 values, using pseudo-moment matrices whose objectives exceed $c_L = 2 + \alpha + \beta$8 (verified via Sturm's theorem). The decay of $c_L = 2 + \alpha + \beta$9 thresholds with level is geometric, certifying the divergence of the required NPA level as $0<\alpha+\beta<2$0.

Geometric Interpretation and Universal Degeneracy

The geometric origin of the phase transition is described by the curvature at the deterministic vertex ($0<\alpha+\beta<2$1): the quantum curve is cubically flat, while every finite-level NPA relaxation remains strictly convex. The limiting obstruction in the scaling limit is the Motzkin polynomial, a classical example of a nonnegative but non-SOS form, revealing the universality and noncommutative character of the certificate growth. The limiting value matches the cubic coefficient of $0<\alpha+\beta<2$2.

Supercritical Phase: Explicit Rational Certificates

For $0<\alpha+\beta<2$3, the paper provides explicit rational PSD matrices $0<\alpha+\beta<2$4 such that $0<\alpha+\beta<2$5 certifies $0<\alpha+\beta<2$6 for all normalized moment vectors $0<\alpha+\beta<2$7 at level $0<\alpha+\beta<2$8. Thus, the NPA hierarchy is exactly tight at all levels in the supercritical quadrant—a nonnegative combination proof requiring no boundary or convexity arguments. The phase transition is sharp, and its location coincides with the boundary where the quadratic coefficient $0<\alpha+\beta<2$9 ceases to be positive.

Dichotomy via Contact Order and Boundary Hessian Condition

The phase transition is theoretically dictated by the boundary Hessian condition. Nondegenerate self-tests (contact order two, non-zero second derivative) guarantee finite-level exactness via Marshall's theorem. The singly-tilted CHSH functional is one such case, where exactness is achieved at level $c_Q(\alpha, \beta)$0. In contrast, the doubly-tilted critical line exhibits a cubic touch (contact order three), where strict complementarity fails and finite-level exactness does not hold, as proven in the companion work. The obstruction is geometric and persists for all finite NPA levels.

Noncommutative Structure and Limitations of Commutative Reduction

The critical obstruction resides in noncommutative moment structure. While commutative reductions (evaluating in the self-tested representation) yield SOS at bounded degree, the divergence in certificate degree is carried exclusively by the noncommutative hierarchy (word length in $c_Q(\alpha, \beta)$1), specifically the Bob-side degree. Asymmetric moment matrices confirm the minimal required Bob length increases as $c_Q(\alpha, \beta)$2.

Errata in Prior Work

Three errata in Gigena et al.'s published polynomial system are documented and corrected, ensuring computational consistency and precision in the reproduced results.

Implications and Future Directions

The results resolve the open question of whether any finite NPA level is exact near the critical doubly-tilted CHSH line: the answer is negative. This establishes a dichotomy in the finite-level semidefinite relaxation landscape for quantum certification, rooted in geometric contact order. Practically, the findings clarify the limits of device-independent certification approaches using the NPA hierarchy and inform the degree bounds necessary for nonlocality certification in critical scenarios. Theoretically, this augments our understanding of spectrahedral shadows and their noncommutative geometry in quantum information.

A quantitative law for the decay of overshoot coefficients $c_Q(\alpha, \beta)$3 and a sharp dimension law for tangency spaces remain open for investigation. The noncommutative character of the obstruction, inaccessible via commutative SOS or generic spectrahedral theory, suggests the necessity to refine dual pseudo-moment techniques and explore noncommutative hierarchy growth in broader scenarios.

Conclusion

The paper rigorously characterizes a phase transition in the exactness of the NPA hierarchy for doubly-tilted CHSH functionals. The transition is geometric, dictated by a contact-order dichotomy and manifesting as a cubic degeneracy on the quantum boundary that provably evades finite-level certification. Explicit rational certificates, sharp analytic expansions, and a universal scaling mechanism are provided, with implications for both quantum information theory and the theory of semidefinite relaxations in noncommutative optimization.