Modular Cocycles, Explicit Class Field Theory, and Quantum Designs

PI和其他人最近的研究进展揭示了数论和量子信息理论中主要开放问题之间的联系。在数论中，希尔伯特的第12个问题（1900年提出的著名的23个问题之一）和相关的斯塔克猜想（在20世纪70年代末形成）关注的是为涉及代数数的某些抽象结构找到更明确的表达式。在量子信息理论中，Zauner猜想（1999）预测了高度规则的几何构型（称为SICs）的存在，它描述了量子测量。该研究项目包括调查希尔伯特/斯塔克问题和Zauner猜想之间的联系，以在这两个领域提供新的见解。该项目的结果将允许更快的sic计算，这在量子态层析成像以及雷达的经典压缩感知方面具有潜在的应用。该项目还将支持研究生的参与，允许PI和他的学生通过会议和研讨会传播他们的研究成果，支持PI所在机构的数论研究研讨会，并支持PI向高中生的推广工作。在该项目中，PI将证明复解析模环的新结果，以改进实二次域的Stark猜想的陈述，并（通过与Appleby和Flammia的联合工作）在每个维度上猜想地构造对称信息完备的正算子值测度（sic - povm或sic）。该项目将发展（广义的，乘法的）解析模共环理论及其“实乘法值”，用类似于模形式的语言而不是l函数的语言重新表述（并证明结果）Stark猜想。本文将通过结构推广sic，对实数二次场给出显式类场论的几何解释。将探讨与Charollois、Dasgupta、Greenberg、Hill、schech和Solomon研究的Eisenstein和Shintani环的联系，以及与Darmon、Pozzi和Vonk研究的p-adic刚性亚纯环的联系。与量子场论的联系也将被探索，数学物理文献中的广义β积分关系将被应用于模环及其实际乘法值。该项目由数学科学部的代数与数论和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Recent research developments by the PI and others have revealed a connection between major open problems in number theory and quantum information theory. In number theory, Hilbert's 12th Problem (one of a famous list of 23 problems proposed in 1900) and the related Stark conjectures (formulated in the late 1970s) concern finding more explicit expressions for certain abstract structures involving algebraic numbers. In quantum information theory, Zauner's conjecture (1999) predicts the existence of highly regular geometric configurations called SICs, which describe quantum measurements. The research project involves investigating a connection between the Hilbert/Stark problems and Zauner's conjecture to provide new insights in both areas. Results from the project will allow for faster computation of SICs, which have potential applications to quantum state tomography as well as classical compressed sensing for radar. The project will also support a graduate student's involvement, allow the PI and his students to disseminate their work through conferences and seminars, support a research seminar in number theory at the PI's institution, and support outreach work by the PI to high school students.In the project, the PI will prove new results on complex analytic modular cocycles to refine the statement of the Stark conjectures for real quadratic fields and (through joint work with Appleby and Flammia) to conjecturally construct symmetric informationally complete positive operator-valued measures (SIC-POVMs or SICs) in every dimension. The project will develop the theory of (generalized, multiplicative) analytic modular cocycles and their "real multiplication values," reformulating (and proving results towards) the Stark conjectures in a language similar to the language of modular forms rather than L-functions. It will produce a geometric interpretation of explicit class field theory for real quadratic fields through structures generalizing SICs. Connections will be explored to Eisenstein and Shintani cocycles as studied by Charollois, Dasgupta, Greenberg, Hill, Sczech, and Solomon and to p-adic rigid meromorphic cocycles as studied by Darmon, Pozzi, and Vonk. Connections to quantum field theory will also be explored, and generalized beta integral relations from the mathematical physics literature will be applied to modular cocycles and their real multiplication values.This project is jointly funded by the Algebra and Number Theory in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.