Quantum Information Meets Mathematics: the Blessing of High Dimension

该项目涉及量子信息科学与数学的接口研究。在过去的几年里，量子信息理论与几何泛函分析（Banach空间理论，算子空间，高维概率）有着非常密切的联系。渐近几何分析，这属于萨雷克博士的专业知识。简而言之，渐近几何分析研究凸集（或其他几何结构）的定量性质及其在维数达到无穷大时的“近似”对称性。这使得它非常适合于量子系统的研究，其中设置是固有的高维。虽然经典的高维现象分析经常受到维数灾难的影响（问题的复杂性随着维数的增加而爆炸，因此问题很快就不再容易处理），但我们可以说渐近几何分析利用了维数的好处，只有当维数很大时，上面提到的对称性才变得明显。Szarek博士在这个项目中的目标是更深入地了解计算机科学和量子物理学的相关方面，这将允许更深入地探索渐近几何分析和量子理论之间的联系。随后的目标包括识别可能有助于解决已知问题的数学技术，以及更一般地，开发与量子信息任务相关的过程相关的数学框架。具体的问题和议题将被研究涉及到可扩展性，PPT属性，量子对象/过程的概率模型，去随机化和对偶的作用。构建量子计算机的探索是21世纪世纪的主要科学和技术挑战之一，量子信息理论提供了理论框架。该领域发展的一个重大事件将是一个为期一个学期的方案“量子信息中的数学挑战”，该方案将于2013年秋季在英国剑桥的艾萨克·牛顿数学科学研究所举行。该计划将汇集理论物理学家，计算机科学家和数学家对该主题感兴趣，并将为具有不同背景的专家之间的跨学科互动提供理想的环境。NSF的资金将支持Szarek博士在牛顿研究所的居住计划期间。虽然Szarek博士是一位理论数学家，但这项研究预计将有助于理解量子信息系统的能力和局限性，因此可能会产生深远的直接和间接影响。此外，该项目将最终导致研究生和本科生参与深入研究，从而促进科学基础和基础设施的发展。

This project involves research on the interface of quantum information science and mathematics. Over the last few years it has become clear that quantum information theory enjoys very close links to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability) a.k.a. asymptotic geometric analysis, which belongs to the expertise of Dr. Szarek. In a nutshell, asymptotic geometric analysis studies quantitative properties of convex sets (or other geometric structures) and their "approximate" symmetries as the dimension goes to infinity. This makes it ideally suited to the study of quantum systems, where the setting is inherently high-dimensional. While classically analyzing high-dimensional phenomena often suffers from the curse of dimensionality (the complexity of the problem exploding with the increase in dimension, so that the question quickly ceases to be tractable), we may say that asymptotic geometric analysis exploits the blessing of dimensionality, with the symmetries mentioned above becoming apparent only when the dimension is large. Dr. Szarek's goals for this project are centered on acquiring a more thorough understanding of relevant aspects of computer science and of quantum physics, which will permit exploring in greater depth the links between asymptotic geometric analysis and the quantum theory. Subsequent objectives include identifying mathematical techniques that may be helpful in solving known problems and, more generally, in developing mathematical framework pertinent to processes associated with quantum information tasks. Specific problems and issues to be studied are related to distillability, the PPT property, probabilistic models for quantum objects/processes, derandomization and the role of duality.The quest to build a quantum computer is one of the major scientific and technological challenges of the 21st century, and quantum information theory provides the theoretical framework for that quest. A major event in the development of that field will be a semester-long program Mathematical Challenges in Quantum Information, to be held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England in the fall of 2013. The program will bring together theoretical physicists, computer scientists and mathematicians with interests in the subject and will supply an ideal environment for an interdisciplinary interaction between experts with different backgrounds. NSF funding will support Dr. Szarek's residency at the Newton Institute for the duration of the program. While Dr. Szarek is a theoretical mathematician, the research is expected to contribute to the understanding of the capabilities and limitations of quantum information systems and thus may have far reaching direct and indirect impact. Additionally, the project will ultimately result in involvement of graduate and undergraduate students in intensive research, thus contributing to the development of scientific base and infrastructure.