Operator Modules, Quantum Groups and Quantum Information

1925年，随着海森堡矩阵力学的出现，我们对物理世界的看法发生了巨大的变化。他证明，通过将时间依赖变量解释为非对易矩阵而不是函数，我们可以准确地描述量子现象。这种函数的“量子化”是量子力学理论发展的基础，促使数学家对其他数学领域进行量子化，包括泛函分析、调和分析和信息论。由此产生的领域：算符空间、拓扑量子群和量子信息理论，至今仍是现代分析和数学物理前沿的世界范围内的重要研究领域。它们都被证明具有深刻的结构理论，它们之间的深刻数学联系不断涌现。我的研究计划是在这三个领域的交汇处进行的。*上述领域相互作用的一个领域是当前量子群调和分析的快速发展。这一理论中感兴趣的算子空间具有非交换代数上的自然模结构，而实际上，相应的算子模结构是该理论的基础。类似于泛函分析的量子化到算子空间的分析理论，我的研究计划的主要长期愿景是发展算子模的分析理论及其在非对易调和分析中的应用。这一发展将为现代泛函分析创造一个全新的方面，在量子群论中具有广阔的应用前景，包括可能解决几个重要的公开问题。该提案中概述的原始技术已经对量子群论产生了重大影响，它们的发展将继续为该理论的发展提供新的工具。*拟议研究的另一个长期目标是开发算符空间和调和分析在量子信息中的新应用。这些领域已经为量子信息理论提供了宝贵的工具，探索它们之间的进一步联系是非常有兴趣的。特别是，我们的目标是探索非对易调和分析与量子纠缠的基本结构之间的最新联系，这与算子代数中最大的公开问题之一密切相关。*这项提议中的跨学科研究，以及我的跨学科背景，为HQP在所有水平上提供了极好的培训机会。这种跨学科的方法促进了HQP在数学上的迅速成熟，以及多才多艺的技能、技术和观点的发展，这为他们在几个相互作用的研究领域内的未来努力提供了特殊的准备。

In 1925 our view of the physical world drastically changed with the advent of Heisenberg's matrix mechanics. He showed that we may accurately describe quantum phenomena by interpreting time dependent variables as non-commuting matrices rather than functions. This "quantization" of functions, which underlies the theoretical development of quantum mechanics, has motivated mathematicians to quantize other areas of mathematics, including functional analysis, harmonic analysis, and information theory. The resulting areas: operator spaces, topological quantum groups, and quantum information theory, are, to this day, prominent world-wide research areas at the forefront of modern analysis and mathematical physics. They have all been shown to have a profound structure theory, and deep mathematical connections between them continue to emerge. My research program lies at the confluence of these three areas.******An area where the above fields interact in a fruitful manner is the current rapid development of harmonic analysis on quantum groups. The operator spaces of interest in this theory carry a natural module structure over a non-commutative algebra, and, in fact, the corresponding operator module structure is fundamental to the theory. Analogous to the quantization of functional analysis to the analytical theory of operator spaces, the main long-term vision of my research program is the development of the analytical theory of operator modules and their applications to non-commutative harmonic analysis. This development will create an entirely new facet of modern functional analysis with promising applications to quantum group theory, including the potential resolution of several important open problems. The original techniques outlined in the proposal have already had a significant impact on quantum group theory, and their development will continue to furnish the theory with novel tools for its evolution.******Another long-term goal of the proposed research is the development of new applications of operator spaces and harmonic analysis to quantum information. These areas have already provided valuable tools for quantum information theory, and it is of great interest to explore further connections between them. In particular, we aim at exploring a recent connection between non-commutative harmonic analysis and the fundamental structure of quantum entanglement, which is intimately related to one of the biggest open problems in operator algebras.******The interdisciplinary research in this proposal, together with my interdisciplinary background, provide excellent training opportunities for HQP at all levels. Such an interdisciplinary approach fosters the rapid mathematical maturity of HQP as well as the development of versatile skills, techniques and perspectives, which provide them with exceptional preparation for future endeavors within several interacting research areas.*****