Correlation and Entanglement in Quantum Systems

我的研究方向是量子系统中的关联和纠缠理论，以及关联和纠缠在量子信息处理和多体物理中所起的作用。纠缠是超越任何可能的经典关联的量子关联，在量子信息论和多体物理的研究中起着关键作用。经过几十年的努力，我们对量子纠缠的理解仍然有限。主要的困难在于希尔伯特空间的维度随着系统中粒子的数量呈指数增长。也就是说，多体量子态需要用指数级的多个复参数来描述。*必须从这些参数中检索能够反映系统的物理意义或信息理论意义的最重要的量，这些量可能具有基本意义或实际意义。通过拟议的研究，我想达到的最终目标是从信息论的角度，为量子系统中的关联和纠缠发展一个一致和统一的理论，并了解它们如何在基本水平上使量子信息处理与经典信息处理不同。*为了实现这一长期目标，我的短期目标包括：理解相同粒子系统(玻色子和费米子)中纠缠的信息论(即可操作性)意义；发展研究边缘重叠的量子边缘问题的方法和条件；了解关联等级(即真正的多体关联/纠缠)及其与拓扑纠缠熵概念的联系；研究约化密度矩阵的几何；并基于不同的Hadamard矩阵，在更一般的情况下理解量子纠错码的结构。*本研究实质上是问题驱动的。也就是说，我会探索任何可能的途径/方法，以便更好地了解问题。因此，我不会局限于任何特定的方法/方法，我将永远愿意学习新的方法。还有一些我将经常使用的典型方法，其中包括：不变量理论、凸几何方法、半定规划、量子德·菲内蒂定理、随机抽样、信息论方法和量子电路方法。这些具体的途径/方法也将使HQP在将它们应用于每个具体项目以实现短期目标时得到很好的培训。*我相信这项工作如果成功，将加强我们对量子系统中的关联和纠缠的理解，以及如何利用它们。**

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. **