Quantum Entanglement and Related Mathematics

该项目致力于系统地发展量子信息和量子计算中一些重要的开放问题的一般理论和某些数学方法。特别是，我们计划研究量子纠缠态的数学结构和物理性质，通过研究局域幺正变换下的不变量，以及张量空间上的算子代数和矩阵分解。我们研究了多体量子态的纠缠度量、可分性的操作条件、量子混合态的等价类以及局域么正变换下轨道的拓扑/几何性质。然后利用多体纠缠态研究在非最大纠缠情况下的最优遥克隆和密集编码。还将研究具有噪声信道的量子密码方案。对于用连续变量描述量子态的情况，应采用随机分析的方法来发展纠缠理论和研究退相干现象。

This project is devoted to a systematic development of the general theory and certain mathematical methods in dealing with some important open problems in quantum information and quantum computation. In particular, we plan to investigate the mathematical structures and physical properties of quantum entangled states, by studying the invariants under local unitary transformations, as well as the operator algebra and matrix decompositions on tensor spaces. We intend to study a measure of entanglement for multipartite quantum states, an operational condition for separability, the equivalence class of quantum mixed states and topological/geometric properties of orbits under local unitary transformations. Multipartite entangled states shall be then exploited to study optimal telecloning and dense coding when the resource used is not maximally entangled. A quantum cryptography scheme with noisy channels shall also be investigated. For the case where the quantum states are described by continous variables, methods of stochastic analysis shall be used to develop an entanglement theory and study decoherence phenomena.