标题： 超越正规矩阵的部分迹关系 显示英文标题

标题： Partial trace relations beyond normal matrices

摘要： 我们研究一般复矩阵的部分迹与其扩张之间的关系，重点关注两个主要方面：(联合)扩张的存在性以及部分迹与其扩张之间的范数不等式。在整个分析过程中，我们特别关注秩约束。我们发现，任何大小相等且迹相同的矩阵对都存在大于一的任意秩的扩张。我们将Audenaert的次可加性不等式推广到涵盖一般矩阵、多个张量因子和不同的范数。这一成果的核心要素是一个关于Kronecker和的新序关系。作为应用，我们将Werner态在其中在任何维度$d\geq4$下可证明为2-不可纯化的区间进行了扩展。我们还证明了新的施密特数见证和$k$-正映射。 显示英文摘要

摘要： We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert's subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably 2-undistillable in any dimension $d\geq4$. We also prove new Schmidt-number witnesses and $k$-positive maps.