多可观测量不确定关系及其在纠缠识别、相干度量与量子网络非局域性中的应用研究

Quantum information is an interdisciplinary research field among Information sciences, Physics and Mathematics. Quantum uncertainty and nonlocality are the fundamental characteristics of quantum mechanics, quantum entanglement and coherence are two important resources of quantum communication, and there exist close links among them. The goal of this research application proposal is to study multi-observable quantum uncertainty relations and their applications on characterizing entanglement criteria and coherence measures in multi-partite continuous variable systems and nonlocality in the quantum network based on operator theory and operator algebras. Specifically, we devote to characterizing multi-observable uncertainty relations based on the sum of variances and the product of standard deviations of observables, and joint uncertainty relations for general non-self-adjoint operators. We will study the topology on the space of self-adjoint operators, which is induced by the standard deviation of operators, and relative topics. Such a pure mathematical problem arises from the research on quantum uncertainty relations. By using uncertainty relations, we shall give a characterization of new criteria for entangled states on multi-partite continuous variables systems. In particular, we investigate simplified entanglement criteria for multi-mode Guassian states, represented by elements of their covariance matrices. We shall characterize the coherence measures of multi-mode Gaussian states and relative open problems based on the quantum uncertainty. We will establish multi-observable entropic uncertainty relations and analyze quantitatively the connection between entropic uncertainty relations and nonlocality in quantum networks. The research in this proposal would not only throw light onto unsolved problems in the quantum information theory, but also highlight the field of operator theory and operator algebras, where further research could be highly illuminating.

量子信息学是信息科学、物理学与数学相交叉的学科.量子不确定性与非局域性是量子力学的基本特性,量子纠缠与相干是量子信息理论中的重要资源,且这四者之间联系紧密.申请项目拟基于算子理论与算子代数研究多可观测量不确定性关系及其在多体连续变量系统纠缠识别、高斯态的相干度量与量子网络中的非局域性等问题中的应用.具体内容包括:计算获得一系列基于标准差乘积与方差和的多可观测量不确定性关系以及一般非自伴算子的不确定性关系,还开展对自伴算子的标准差诱导的拓扑性质及相关基础数学问题的研究;基于不确定关系给出多体连续变量系统态(特别是多模高斯态)的更优的纠缠判据;利用量子不确定性研究高斯态相干度量及相关问题;建立某些量子网络中多可观测量熵不确定性关系,通过定量计算揭示其与相应量子网络非局域性之间的联系.申请项目课题研究成果不仅回答量子信息学未解决问题,也提出和推动算子理论与算子代数的新课题与发展.

量子信息学是一门数学、物理学与计算机科学相交叉的学科，已成为当今最热门的研究领域之一。近年来算子理论与算子代数学者从自身学科优势出发去研究量子信息学中的问题,已成为这个研究领域的新亮点。量子不确定性与量子关联是量子信息学中的基本概念和核心内容，而二者之间又存在密切的联系。该结题项目旨在从算子理论角度出发获得系列多可观测量不确定性关系，并研究其在纠缠识别、相干度量与量子网络非局域性理论中的应用。具体的，获得了一类任意N个可观测量不确定性关系，证明该不确定性关系是紧凑的，在N=2时是比原始的海森堡、罗伯逊不确定性关系更优，与薛定谔不确定性关系等价；获得了基于斜信息表示的两个连续投影测量的不确定性关系；利用不确定性关系构造了一类高斯态纠缠判据，这个判据的优势在于只要从一个高斯态的相关矩阵中的元素不等式关系就可以判断这个高斯态的纠缠性；刻画了高斯破坏相干信道，利用这一刻画，构造了一类高斯相干度量，进一步研究了测量诱导平均总量子相干性的互补性关系；获得了一类存在量子记忆的多可观测量熵不确定性关系，并给出了一个紧凑的下界；得到一类马尔可夫过程中的熵不确定性关系；获得了两类纠缠swapping网络中量子非局域性的度量；对于量子门分离与量子可导引性机器学习方法识别进行了初步研究。项目基于算子理论技巧解决了量子信息理论中的系列问题，揭示了量子不确定性与量子关联的新关系，丰富了算子理论与量子信息学两个领域的研究。

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