Certifying Optimality of Bell Inequality Violations: Noncommutative Polynomial Optimization through Semidefinite Programming and Local Optimization
本文从非交换多项式优化的角度研究贝尔不等式及其违反，利用NPA层级、GNS构造和牛顿芯片技术等方法，认证了标准贝尔不等式集合中大部分不等式违反的最优性。
Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large majority of a set of standard Bell inequalities, denoted A2-A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton-McCullough Positivstellensatz,the Gelfand-Naimark-Segal (GNS) construction with a novel use of the Artin-Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new "Newton chip"-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented.This technique is based on conditional expectations. Finally, noncommutative Gröbner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution.