Grothendieck常数、量子相关矩阵与CCP函数的上界
Upper bounds for Grothendieck constants, quantum correlation matrices and CCP functions
AI总结：
本文基于Hadamard型相关矩阵映射函数与高斯、积分及级数反演工具，提出统一推导实复Grothendieck常数上界的框架，恢复并简化经典最强界。
摘要：
在著名Grothendieck不等式中实数与复数Grothendieck常数$K_G^\mathbb{F}$的确切值仍未知（自1953年以来尚未解决）的探索框架下，其中$\mathbb{F}$表示实数域或复数域，我们将搜索集中于其最小上界。为此，我们建立了一个基础框架，该框架构建于通过Hadamard乘积逐元素将相关矩阵映射到相关矩阵的函数之上，例如实数情形中的Krivine函数或复数情形中的Haagerup函数。利用多元实高斯与复高斯分析、高等超越函数、球面积分以及Maclaurin级数反演的组合数学，我们给出一种方法，并借此恢复Grothendieck本人（$K_G^\mathbb{R} \leq \sinh(π/2) \approx 2.301$）、Krivine（$K_G^\mathbb{R} \leq \fracπ{2 \ln(1 + \sqrt{2})} \approx 1.782$）和Haagerup（$K_G^\mathbb{C} \leq 1.405$，数值近似）的所有著名上界；每一个都作为特殊情形。在此过程中，我们旨在尽可能统一实数与复数情形，并将结果应用于若干具体例子，包括Walsh-Hadamard变换（“量子门”）和多元Gaussian copula——同时考虑量子理论与量子信息论的基础。此外，我们缩短并简化了迄今最强估计的证明；即$K_G^\mathbb{R} < \fracπ{2 \ln(1 + \sqrt{2})}$。我们以算法方案的形式总结关键结果，并阐明相关开放问题与未来研究主题。
英文摘要：
Within the framework of the search for the still unknown exact value of the real and complex Grothendieck constant $K_G^\mathbb{F}$ in the famous Grothendieck inequality (unsolved since 1953), where $\mathbb{F}$ denotes either the real or the complex field, we concentrate our search on their smallest upper bound. To this end, we establish a basic framework, built on functions which map correlation matrices to correlation matrices entrywise by means of the Hadamard product, such as the Krivine function in the real case or the Haagerup function in the complex case. By making use of multivariate real and complex Gaussian analysis, higher transcendental functions, integration over spheres and combinatorics of the inversion of Maclaurin series, we provide an approach by which we also recover all famous upper bounds of Grothendieck himself ($K_G^\mathbb{R} \leq \sinh(π/2) \approx 2.301$), Krivine ($K_G^\mathbb{R} \leq \fracπ{2 \ln(1 + \sqrt{2})} \approx 1,782$) and Haagerup ($K_G^\mathbb{C} \leq 1.405$, numerically approximated); each of them as a special case. In doing so, we aim to unify the real and complex case as much as possible and apply our results to several concrete examples, including the Walsh-Hadamard transform (''quantum gate'') and the multivariate Gaussian copula - with foundations of quantum theory and quantum information theory in mind. Moreover, we offer a shortening and a simplification of the proof of the strongest estimation until now; namely that $K_G^\mathbb{R} < \fracπ{2 \ln(1 + \sqrt{2})}$. We summarise our key results in form of an algorithmic scheme and shed light on related open problems and topics for future research.
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