论文标题
对痕量多项式的优化
Optimization over trace polynomials
论文作者
论文摘要
由量子信息理论的最新进展激励，本文旨在优化痕量多项式，即在非交通变量和产品痕迹中的多项式。提出了一种新型的阳性态度，证明了痕量多项式受到痕量约束的阳性，并提供了单调地收敛至最佳痕量多项式受试者对奇特限制的最佳降低的层次结构。该层次结构可以看作是Pironio，Navascués和Acín方案[New J. Phys。，2008年]的奇特类似物，以优化非交通性多项式。如果满足平稳性和极端条件，则使用Gelfand-Naimark-segal（GNS）结构来提取痕量优化问题的优化者。这些条件足以获得我们层次结构的有限收敛。获得的结果应用于量子信息理论中对多项式钟的不平等现象的侵犯。本文中使用的主要技术的灵感来自实际代数几何，操作者理论和非交通代数。
Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme [New J. Phys., 2008] for optimization of noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.
