发表机构
Key Laboratory of System Software (Chinese Academy of Sciences), Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; Department of Computer Science and Technology, Tsinghua University（中国科学院软件研究所系统软件重点实验室; 中国科学院大学; 清华大学计算机科学与技术系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否定回答了Tsirelson关于量子行为集合是否为半代数集的问题，证明在每方四个二元测量下，有限维量子行为集合、其闭包及算子交换集合均非半代数，且局部无有限实解析描述，排除了精确有限半定表示和有限NPA层级刻画。
AI 中文摘要
在贝尔场景中，对共享量子态进行局域测量所能实现的条件概率构成量子行为集合，其结构由Tsirelson研究。1993年，Tsirelson提出该集合是否为半代数集的问题，即能否用多项式方程与不等式的有限布尔组合来描述。此问题一直悬而未决。我们给出否定答案：对于每方四个二元测量，有限维量子行为集合、其闭包以及算子交换集合均非半代数。更强地，这些集合在某个特定经典行为附近甚至局部都不具有有限实解析描述。这些结果排除了精确的有限半定表示，并表明Navascués-Pironio-Acín层级中没有任何有限层级能精确刻画这些集合。
英文摘要
The conditional probabilities achievable by local measurements on a shared quantum state in a Bell scenario form the set of quantum behaviors, whose structure was studied by Tsirelson. In 1993, Tsirelson asked whether this set is semialgebraic, that is, describable by finite Boolean combinations of polynomial equations and inequalities. This question has remained open. We answer it in the negative: with four binary measurements per party, the set of finite-dimensional quantum behaviors, its closure, and the commuting-operator set are all nonsemialgebraic. More strongly, none admits a finite real-analytic description even locally near a particular classical behavior. These results rule out exact finite semidefinite representations and show that no finite level of the Navascués-Pironio-Acín hierarchy characterizes these sets exactly.
