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

Research Question and Approach

This paper investigates the exactness of the Navascués-Pironio-Acín (NPA) hierarchy when applied to the doubly-tilted CHSH functional, defined as B_αβ = α⟨A0⟩ + β⟨B0⟩ + CHSH. While previous work established the quantum maximum for this functional, it remained an open question whether any finite level of the NPA hierarchy could reach this maximum near the critical line α + β = 2. The author quantifies the hierarchy's performance by analyzing the "overshoot"—the difference between the NPA level-k value and the true quantum maximum—as a function of the tilt parameter s = 2 - α - β.

The study identifies a phase transition in the hierarchy's exactness. In the supercritical region (α, β ≥ 1), the author provides explicit rational certificates that prove the hierarchy is exact at every level. Conversely, in the subcritical region, the hierarchy fails to be exact at any finite level near the critical line. The author demonstrates that the required NPA level to reach the quantum maximum diverges as the tilt approaches the critical line. This failure is linked to the Motzkin polynomial, a classical example of a non-negative form that is not a sum of squares, indicating that the limitation is structural rather than a result of insufficient degree.

This work resolves a significant open problem regarding the convergence of the NPA hierarchy in quantum information theory. By identifying the exact mechanism of failure—a cubic-touching maximum at the critical vertex—the paper provides a precise geometric interpretation of why certain quantum optima are difficult to certify. The documentation of verified errata in previous literature and the provision of exact rational certificates for the supercritical phase offer a robust foundation for future research into the limits of semidefinite programming relaxations in quantum Bell scenarios.