"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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=596285
• 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).

近年来，随机结构已成为量子信息理论（QIT）中非常富有成果的工具。高维设置在QIT中非常普遍，因为QIT的数学描述（以及其他科学和工程）问题通常涉及大量自由度（可以解释为维度）。例如，8 Qutrit量子系统的状态空间的维度超过4300万。这样的高维设置使数值方法不切实际。但是，由于“中心极限定理”效应，几何功能分析（GFA）和随机矩阵（RM）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，并且随着尺寸变大，矩阵有价值的概率措施。 QIT的另一个常见特征是凸度。 QIT中许多感兴趣的对象是凸体。因此，了解凸体的几何特性对于QIT很重要，并且是凸几何分析（CGA）的主要目标。