Operator Modules, Quantum Groups and Quantum Information

算子模块、量子组和量子信息

• 批准号: RGPIN-2017-06275
• 负责人: Crann, Jason
• 金额: $ 1.53万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=648255
• 关键词: Operator; Modules; Quantum; Groups; Information

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.*****

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。他表明，我们可以通过将时间相关变量解释为非交换矩阵而不是函数来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，它激励数学家将数学的其他领域量子化，包括泛函分析、谐波分析和信息论。由此产生的领域：算子空间、拓扑量子群和量子信息论，至今仍是现代分析和数学物理前沿的突出的全球研究领域。它们都被证明具有深刻的结构理论，它们之间深刻的数学联系不断出现。我的研究项目就在这三个领域的交汇处。******上述领域相互作用的一个富有成效的领域是当前量子群谐波分析的快速发展。该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，事实上，相应的算子模结构是该理论的基础。与泛函分析的量化与算子空间的解析理论类似，我的研究计划的主要长期愿景是发展算子模块的解析理论及其在非交换谐波分析中的应用。这一发展将为现代泛函分析创造一个全新的方面，并有望应用于量子群论，包括解决几个重要的开放问题。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量子群论的进化提供新的工具。******提出的研究的另一个长期目标是开发算子空间和谐波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有价值的工具，探索它们之间的进一步联系是非常有趣的。特别是，我们的目标是探索非对易调和分析和量子纠缠的基本结构之间的联系，这与算子代数中最大的开放问题之一密切相关。******本次提案的跨学科研究，加上我的跨学科背景，为HQP的各个层次提供了极好的培训机会。这种跨学科的方法促进了HQP的快速数学成熟，以及多功能技能，技术和观点的发展，为他们在几个相互作用的研究领域的未来努力提供了特殊的准备。*****