Quantum Entanglement and Related Mathematics

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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.