论文标题
通过随机测量探测相关矩阵的几何形状
Probing the geometry of correlation matrices with randomized measurements
论文作者
论文摘要
两分量子态的广义BLOCH分解产生了相关矩阵，其奇异值提供了有关状态非本地性质的丰富信息，例如纠缠的维度。尽管存在一些基于单数值的纠缠标准，但缺乏对可接受相关矩阵的几何形状的完全理解。我们对有限施密特数字的相关矩阵的奇异值的几何形状提供了更深入的见解。首先，我们通过构造与从Bloch球体上的正交平均值相同的时刻来构造相同时刻的可观察到的物体来提供指向随机测量框架的链接，并展示如何在此框架中获取有关奇异值的知识。然后，我们将重点放在可分离状态的情况下，并通过为某些面部和极端点提供明确的构造来表征前两个非变化矩的边界。这些构造产生了相关矩阵的几何形状与相互无偏基的最大集合的存在问题以及SIC-POVM之间的联系。
The generalized Bloch decomposition of a bipartite quantum state gives rise to a correlation matrix whose singular values provide rich information about non-local properties of the state, such as the dimensionality of entanglement. While some entanglement criteria based on the singular values exist, a complete understanding of the geometry of admissible correlation matrices is lacking. We provide a deeper insight into the geometry of the singular values of the correlation matrices of limited Schmidt number. First, we provide a link to the framework of randomized measurements and show how to obtain knowledge about the singular values in this framework by constructing observables that yield the same moments as one obtains from orthogonal averages over the Bloch sphere. We then focus on the case of separable states and characterize the boundary of the set of the first two non-vanishing moments by giving explicit constructions for some of the faces and extremal points. These constructions yield a connection between the geometry of the correlation matrices and the existence problems of maximal sets of mutually unbiased bases, as well as SIC-POVMs.
