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 $B_{\alpha \beta }=\alpha ⟨A_0⟩+\beta ⟨B_0⟩+\mathrm{CHSH}$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $\alpha +\beta =2$. We quantify the mechanism on the symmetric slice $s=2-\alpha -\beta$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k{s}^2+O(s^3)$, 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 $\alpha ,\beta \ge 1$ 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.

11 pages, Companion to "No finite level of the NPA hierarchy is exact for the doubly-tilted CHSH functional near the critical tilt", submitted simultaneously. Certificates and verification code at https://github.com/tohafrit/npa-nonexactness