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

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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.