Analytic Qubit Separation between POVMs and Projective Measurements

This paper addresses a fundamental question in quantum information: can generalized measurements (POVMs) generate quantum correlations that cannot be reproduced by projective measurements (PVMs) when the local Hilbert space dimension is constrained? While Naimark's theorem states that any POVM can be implemented as a PVM in a larger Hilbert space, the operational distinction becomes significant when auxiliary dimensions are unavailable.

The authors design a specific Bell functional that combines a standard CHSH term (the anchor) with a smaller ternary term (the probe). By using a rational deformation of a three-ray qubit measurement, they create a scenario where the POVM advantage is first-order in the probe strength, while the optimal projective compensation is only second-order. They provide a rigorous analytic proof that this POVM strategy outperforms all possible qubit-projective strategies, including those involving shared classical randomness or postprocessing. Additionally, they use a level-3 noncommutative sum-of-squares (SOS) certificate to prove that their explicit strategy is globally optimal among all finite-dimensional quantum strategies.

The study establishes that there exists an explicit behavior generated by a qubit POVM that cannot be simulated by any convex combination of qubit-projective strategies. The certified gap is greater than 1/1000, and the authors provide a formal Lean 4 proof to ensure the correctness of this separation. Furthermore, they demonstrate that their chosen POVM strategy is not just superior to projective measurements, but is the absolute maximum for the defined Bell functional across all finite-dimensional quantum systems.