量子不确定性原理与量子关联的研究

Quantum information theory and its basic problems related to quantum mechanics is one of the research focuses in physics, and have received great attention. Quantum uncertainty principle is one of the essential features of quantum mechanics, and have many useful applications in detecting quantum entanglement and investigation of error-disturbance relations. Correlations, as valuable resources in quantum processing, play a vital role in quantum information theory. Characterization and measurement of correlation are also hot issues in studying quantum information. In this project, we first investigate uncertainty and reverse uncertainty relations for the product as well as the sum of variances of arbitrary finite quantum mechanical observables, and extend the results to non-Hermitian operators, weak values and general unitary operators. Moreover, we study quantum correlation measures and provide a new measure based on skew information. We also combine quantum uncertainty of non-Hermitian operators and quantum correlations, and provide the results having obvious physical meaning. These results have great theoretical significance and potential value.

量子信息理论及其所涉及的量子力学基本问题是近年来物理学的研究热点之一，受到了人们广泛的关注。量子不确定性原理不仅是量子力学区别于经典力学的本质特征之一，而且在量子纠缠检测、误差干扰关系等方面具有重要应用。关联在量子信息中扮演重要角色，它是量子信息处理中非常珍贵的资源。对于关联的刻画与度量一直是量子信息研究的热点问题。本项目首先对于任意多个可观测量研究基于方差的量子不确定性关系以及反向不确定性关系，并且把相关结果推广到非厄米算子、弱值和一般幺正算子的情形。此外，我们研究量子关联的刻画，给出基于斜信息的关联的度量。我们还将尝试把非厄米算子的不确定性关系和关联结合在一起，给出具有明显物理意义的结果。这些结果将对量子信息和量子力学基本问题具有重要的理论意义和潜在的应用价值。

量子纠缠是量子力学的一个基本特征，并且在量子信息处理中扮演重要角色。纠缠的单配性与多配性问题是近年来量子信息理论的研究热点。本项目首先对于任意有限维量子系统给出了一类新的多体纠缠权多配性不等式，然后对于多比特纠缠的并发度、纠缠形成、负度、Tsallis-q纠缠以及Rényi-α纠缠，我们分别给出了几类单配性与多配性不等式。此外，我们研究了互补测量在量子信息中的应用。我们对于一组完备的互补测量计算了方差的加和，并通过分析发现对于此类完备的互补测量，Brukner–Zeilinger不变信息在数值上等于最大方差与总方差的差值。进一步我们分别对于任意一组完备的互无偏测量与一般SIC测量计算了量子态的平均相干度。我们还研究了基于metric adjusted斜信息的互补测量诱导的量子不确定性，利用这类不确定性度量给出了新的纠缠判据，并通过具体例子来说明新的判据在纠缠检测方面更加有效。