$I_{3322}$ 贝尔不等式的量子上确界在有限维中不可达到
The quantum supremum of the $I_{3322}$ Bell inequality is not attained in finite dimension
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中文总结 AI 辅助
本文证明了$I_{3322}$贝尔不等式的量子上确界为有限维策略收敛的极限，且有限维策略无法达到该上确界，有限维量子关联集在对应最小贝尔场景中不闭合，证明核心已在Lean 4中形式化。
中文摘要 AI 辅助
2010年，Pál和Vértesi发现了针对$I_{3322}$贝尔不等式的一系列有限维策略，其优化值随局域希尔伯特空间维度增大似乎收敛。他们推测该极限是所有有限维量子策略的上确界，但不存在有限维策略能达到它。我们证明了这两个论断。该证明利用贝尔泛函的对称性，将每个策略与一个有限概率矩阵关联，该矩阵对应Alice和Bob的谱子空间对，给出贝尔值的上界，且由Pál和Vértesi发现的重复结构构建的有限维策略随维度增大趋近该上界。若该界在有限维中被精确达到，则最优性条件要求存在无法归一化的态。因此，在$(3,3,2,2)$场景中，有限维量子关联集不闭合，这是最小的贝尔场景中会出现此情况的情形，且趋近上确界需要无界限局域维度。证明的核心部分已在Lean 4中形式化。
英文摘要
In 2010, Pál and Vértesi found a family of finite-dimensional strategies for the $I_{3322}$ Bell inequality whose optimized values appeared to converge as the local Hilbert-space dimension grew. They conjectured that this limit is the supremum over all finite-dimensional quantum strategies, but that no finite-dimensional strategy attains it. We prove both claims. The proof uses the symmetry of the Bell functional to associate every strategy with a finite matrix of probabilities, one for each pair of spectral subspaces of Alice and Bob. This matrix gives an upper bound on the Bell value, and finite-dimensional strategies built from the repeating structure found by Pál and Vértesi approach it as the dimension grows. If the bound were attained exactly in finite dimension, the optimality conditions would then require a state that cannot be normalized. Consequently, the set of finite-dimensional quantum correlations is not closed in the $(3,3,2,2)$ scenario, the smallest Bell scenario where this can happen. Moreover, approaching the supremum requires unbounded local dimension. The core of the proof was formalized in Lean~4.
发表机构
- University of Geneva（日内瓦大学）
- Constructor University（Constructor大学）
机构由 AI 辅助整理，请以论文原文为准。
