量子边界问题中秩与特征值的研究

Many important and interesting questions arise with the development of quantum information theory. In this project, we concern with a class of subproblem in quantum marginal problems...In two-partite systems, we find the range of rank of the global states when the two marginal states are given, and characterize the relation of their rank. For given marginal states, if the global states are not pure, discuss closeness between global and pure states by the relation of their eigenvalues. On this basis, study the related conditions for multipartite systems. Investigate the applications of above results in quantum information theory, such as find new methods to characterize the entanglement of quantum states, obtain new entropy inequalities benefited from the method of group theory, etc.

随着量子信息论的发展, 产生了一系列重要而有趣的问题. 本项目关注的是量子边界问题中的一类子问题. ..在两体系统中, 对给定的两个边界态, 本项目研究了其整体态秩的取值范围,以及它们之间秩的关系. 在给定边界态的情况下, 当整体态中不存在纯态时, 利用它们之间特征值的关系刻画整体态与纯态的接近程度. 在此基础上研究多体情况下的相关结果. 并给出上述结论在量子信息论中的应用, 如寻找利用边界态的性质来刻画整体态纠缠性的新方法, 以及通过群论的方法得到新的熵不等式等.