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

近年来，随机结构已成为量子信息论(QIT)中一个卓有成效的工具。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的自由度(可以解释为维度)。例如，一个8qutrit量子系统的态空间的维度超过4300万。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)的“中心极限定理”效应带来了“维度的加持”。这些领域的目的是了解几何对象和矩阵值概率度量的典型特征，随着维度变大。QIT的另一个共同特征是凸性。QIT中的许多感兴趣的物体都是凸体。因此，了解凸体的几何性质对于QIT是非常重要的，也是凸几何分析(CGA)的主要目标。在CGA中，PI将继续他的工作，了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，如PDE、几何层析和信息论。在QIT中，PI将重点了解纠缠、PPT和随机诱发态的性质，并进一步探索QIT、RM、CGA和GFA之间的联系。希望本文的工作将加深对仿射不变量(特别是仿射曲面面积)的理解，有助于为Orlicz-Brunn-Minkowski理论奠定基础，并发展出非实欧氏空间的新的仿射不变量。在量子纠缠研究中，人们期望所提出的工作将为量子纠缠提供更深层次的认识，并为量子纠缠的探测提供有用的结果。最后，拟议的工作将有利于HQP的培训，因为PI有具体的计划让学生参与他的研究。

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.