"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"

“凸几何分析、随机矩阵及其在量子信息论中的应用”

• 批准号: 418296-2012
• 负责人: Ye, Deping
• 金额: $ 1.24万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635268
• 关键词: Convex; Geometric; Analysis; Random; Matrices

In recent years, random constructions have become a very fruitful tool in Quantum Information Theory (QIT). The high-dimensional setting is very common in QIT since mathematical descriptions of QIT (as well as other scientific and engineering) questions often involve a large number of degrees of freedom (which can be interpreted as the dimension). For instance, the state space of an 8 qutrit quantum system is of dimension of more than 43 million. Such a high-dimensional setting makes the numerical approach impractical. However, Geometric Functional Analysis (GFA) and Random Matrices (RM) bring the "blessing of dimensionality" because of the "central limit theorem-like" effects. These areas aim to understand typical features of geometric objects and matrix valued probability measures as the dimension become large. Another common feature for QIT is convexity. Many objects of interest in QIT are convex bodies. Hence, understanding the geometric properties of convex bodies is important for QIT, and is the main goal for Convex Geometric Analysis (CGA).The proposed research will involve several projects in CGA and in QIT. In CGA, the PI will continue his work on understanding properties of affine invariants associated with convex bodies and exploring their connections with other fields, such as PDE, Geometric Tomography, and Information Theory. In QIT, the PI will focus on understanding properties of entanglement, PPT and random induced states, and further explore the connections between QIT, RM, CGA and GFA. It is hoped that the proposed work will deepen the understanding of affine invariants (especially affine surface areas), help build foundation on the Orlicz-Brunn-Minkowski theory, and develop new affine invariants for spaces other than real Euclidean space. In QIT, it is expected that the proposed work will provide much deeper view of quantum entanglement, and provide useful results for detecting quantum entanglement. Lastly, the proposed work will be advantageous to the training of HQP as the PI has specific plans to involve students in his research.

近年来，随机构造已成为量子信息论（QIT）中非常富有成效的工具。高维设置在 QIT 中非常常见，因为 QIT（以及其他科学和工程）问题的数学描述通常涉及大量自由度（可以解释为维度）。例如，8量子系统的状态空间的维数超过4300万维。如此高维的设置使得数值方法不切实际。然而，几何泛函分析（GFA）和随机矩阵（RM）由于“类中心极限定理”效应而带来了“维度的祝福”。这些领域旨在了解几何对象的典型特征以及随着维度变大而进行的矩阵值概率度量。 QIT 的另一个共同特征是凸性。 QIT 中感兴趣的许多物体都是凸体。因此，理解凸体的几何性质对于QIT很重要，也是凸几何分析（CGA）的主要目标。所提出的研究将涉及CGA和QIT中的多个项目。在 CGA 中，PI 将继续致力于理解与凸体相关的仿射不变量的性质，并探索它们与其他领域的联系，例如偏微分方程、几何断层扫描和信息论。在QIT中，PI将重点了解纠缠、PPT和随机诱导态的性质，并进一步探索QIT、RM、CGA和GFA之间的联系。希望所提出的工作能够加深对仿射不变量（特别是仿射曲面区域）的理解，帮助奠定 Orlicz-Brunn-Minkowski 理论的基础，并为实欧几里得空间以外的空间开发新的仿射不变量。在 QIT 中，预计所提出的工作将为量子纠缠提供更深入的视角，并为检测量子纠缠提供有用的结果。最后，拟议的工作将有利于 HQP 的培训，因为 PI 有具体计划让学生参与他的研究。