Correlation and Entanglement in Quantum Systems

我提出的研究方向是量子系统中的相关和纠缠理论，以及相关和纠缠在量子信息处理和多体物理中的作用。

My proposed research is the theory of correlation and entanglement in quantum systems, and the role correlation and entanglement play in quantum information processing and many-body physics. Entanglement, the quantum correlation beyond any possible classical correlation, plays the key role in the study of quantum information theory and many-body physics. After decades of effort, our understanding of quantum entanglement remains limited. The major difficulty lies in the fact that the dimension of the Hilbert space grows exponentially with the number of particles in the system. That is, a many-body quantum state needs to be described by exponentially many complex parameters. It is essential to ‘retrieve’ from those parameters the most important quantities that capture the physical, or information-theoretic meanings of the systems, which could be of either fundamental or practical relevance. The ultimate goal I want to achieve via the proposed research is to develop a consistent and unifying theory for correlation and entanglement in quantum systems, from the information-theoretic viewpoints, and understand how they make quantum information processing different from its classical counterparts at the fundamental level. In order to achieve this long-time goal, my short-time objectives include the following: understand the information-theoretic (i.e. operational) meanings for entanglement in identical particle systems (bosons and fermions); develop methods and conditions to study the quantum marginal problem with overlapping marginals; understand the correlation hierarchy (i.e. ‘true’ many-body correlation/entanglement) and its connection to the concept of topological entanglement entropy; study the geometry of reduced density matrices; and understand the structure of quantum error-correcting codes under a more general setting, based on different Hadamard matrices. The proposed research is essentially ‘problem driven’. That is, I will explore any possible approach/method in order to understand the problem better. Therefore I will not restrict myself to any particular approach/method, and I will be always willing to learn new ones. There are also typical approaches/methods that I will usually use, which include the following: invariant theory, method of convex geometry, semi-definite programing, quantum de Finetti’s theorem, random sampling, information-theoretic methods, and the method of quantum circuits. These concrete approaches/methods will also give HQPs good training when applying them to each specific project toward achieving the short-term objectives. I believe this work, if successful, will strengthen our understanding of correlation and entanglement in quantum systems, and how to make use of them.