量子信息与量子计算是综合了物理学中凝聚态物理和量子力学、计算机科学和数学中的Hopf代数及有限张量范畴理论、量子群及其拓扑量子场论等多学科的新型交叉学科领域，是目前国家科技发展的优先领域。本项目拟用李代数权表示理论、辫子群酉表示理论、模块化张量范畴理论等现代数学工具来研究量子信息和拓扑量子计算。首先将学习范畴化量子力学与拓扑量子计算，试图弄清楚辫子群酉表示和模块化张量范畴在拓扑量子计算中的数学机理、解决Rowell-Wang在Bulletin AMS上提出的关于辫子群酉表示在量子计算方面的一些开问题。同时，用权表示论给出基于方差的和不确定性，并将证明其与量子态的选取无关。另外，也将考虑利用特殊酉李代数的性质优化已有的判断量子态是否纠缠的自旋压缩不等式，并将其推广。

Quantum information and quantum computation are new interdisciplinary fields that integrate physics such as condensed matter physics and quantum mechanics, computer science, and mathematics such as finite tensor categories in Hopf algebra theory and topological quantum field theory arising from quantum groups and knots, links or 3-manifold invariants. It is currently the priority area of national scientific and technological development. This project intends to use modern mathematical tools such as the weight representation theory of Lie algebras, the unitary representation theory of braid groups, and the modular tensor category theory to study quantum information and topological quantum computation. Firstly, through learning the categorifization quantum mechanics and topological quantum computation, we willtry to understand the mathemtical mechanism of the unitary representation of braid group and the modular tensor category in topological quantum computing, and aim to solve some open questions on unitary representations of braid group and their applications in quantum computation proposed by Rowell and Zhenghan Wang in Bulletin AMS. At the same time, the weight representation theory of Lie algebras is used to give the vaiance-based sum uncertainty of semisimple Lie algebra, and it will be proved that it is independent of the choices of quantumstates. In addition, we will also consider using the properties of special unitary Lie algebrasto optimize the existing spin squeezing inequalities for judging whether a quantum state is entanged or not, and extend it.