微分几何和表象理论在量子关联描述中的应用
Applications of differential geometry and representation theory to description of quantum correlations
摘要：
量子信息理论中最重要的问题之一就是所谓的可分性问题。它涉及到描述多体量子系统的混合态中的可分离（或等价纠缠）态的集合。在这篇论文中，我们研究了这个问题的推广类型的量子关联的定义类似于纠缠。我们从给定量子系统的纯态集合的子集开始，并将属于该子集的凸包的状态称为“非相关”状态。因此，位于凸包外部的状态被称为"相关"。在这项工作中，我们专注于当存在一个对称群，保持类的“非相关”纯态的情况下。对称性的存在允许获得对许多类型的看似不相关的相关性的统一处理。我们将我们的一般结果应用于特定类型的相关性：（i）可区分粒子的纠缠，（ii）玻色子的粒子纠缠，（iii）费米子的"纠缠"，（iv）费米子系统中的非凸高斯相关性，（v）真正的多方纠缠，最后（vi）基于施密特数概念的二分纠缠的精炼概念。我们研究自然的问题和问题的相关性定义以上：（一）我们提供明确的多项式表征的各种类型的相关性的纯态。(II)我们研究的情况下，它是可能的相关混合态给出一个完整的分析表征。(III)我们推导出各种多项式的标准检测相关性的混合状态。(IV)我们利用上述准则和浓度测度技术研究了等谱密度矩阵集上相关的典型性质。
英文摘要：
One of the most important questions in quantum information theory is the so-called separability problem. It involves characterizing the set of separable (or, equivalently entangled) states among mixed states of a multipartite quantum system. In this thesis we study the generalization of this problem to types of quantum correlations that are defined in a manner analogous to entanglement. We start with the subset of set of pure states of a given quantum system and call states belonging to the convex hull of this subset "non-correlated" states. Consequently, the states laying outside the convex hull are referred to as "correlated". In this work we focus on cases when there exist a symmetry group that preserves the class of "non-correlated" pure states. The presence of symmetries allows to obtain a unified treatment of many types of seemingly unrelated types of correlations. We apply our general results to particular types of correlations: (i) entanglement of distinguishable particles, (ii) particle entanglement of bosons, (iii) "entanglement" of fermions, (iv) non-convex-Gaussian correlations in fermionic systems, (v) genuine multiparty entanglement, and finally (vi) refined notions of bipartite entanglement based on the concept of the Schmidt number. We investigate the natural problems and questions concerning the correlations defined above: (I) We provide explicit polynomial characterization of various types of correlations for pure states. (II) We examine cases in which it is possible to give a complete analytical characterization of correlated mixed states. (III) We derive a variety of polynomial criteria for detection of correlations in mixed states.(IV) We use the above criteria and the technique of measure concentration to study typical properties of correlations on sets of isospectral density matrices.
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