对称的量子纠缠纯态的理论构造问题

As for the preparation theory of the quantum entangled pure state, there are a large number of results with small particles, but little for the multiple particles and high complexity. On the other hand, compared with lots of the fast decompositions for the sparse matrix, there stills little research for the special symmetric sparse matrix tensor decomposition or approximate decomposition. However, in the quantum information theory, many quantum systems have much special symmetric structures, such as these invariable entangled states under the permutation group or its special subgroups. Therefore, their preparations in theory are very meaningful. This project studies the tensor decomposition of the sparse matrix with some symmetrical structures by using sparse QR decomposition, Cholesky decomposition, Cosine-Sine decomposition sparse quantum etc., and with the aid of quantum information storage theory to construct multi particle entangled pure states.

关于量子纠缠纯态的理论构造问题，目前已有大量结果讨论少粒子情况，然而针对多粒子的结果不多，复杂度也相当高。另一方面，针对稀疏矩阵的快速分解目前已有较多成果，而针对具有特殊对称结构的稀疏酉矩阵的张量分解或近似分解研究还极少。但在量子信息理论中，很多量子系统都具有特殊对称结构，如在置换群或其特殊子群下不变的纠缠态等，因而，其理论构造非常有意义。本项目通过稀疏QR分解、稀疏Cholesky分解、量子Cosine-Sine分解等研究具有对称结构的稀疏酉矩阵的张量分解，进而，借助量子信息存储理论构造多粒子纠缠纯态。

本项目研究特殊对称量子纠缠系统的制备理论及其各种应用。从所取得的成果按照内容可分为三大类：（一）纠缠理论算法的研究，包括可多项式制备的量子纠缠，一般量子纠缠制备的指数复杂度，随机量子纠缠变换。（二）特殊量子纠缠制备实验模型，包括n光子2n 量子比特态的超纠缠纯化。（三）基于纠缠态的各种量子通信应用，包括量子信息的远程重构和分享等。