Title: On decomposable correlation matrices Show Chinese title

Title: 关于可分解相关矩阵

Abstract: Correlation matrices (positive semidefinite matrices with ones on the diagonal) are of fundamental interest in quantum information theory. In this work we introduce and study the set of $r$-decomposable correlation matrices: those that can be written as the Schur product of correlation matrices of rank at most $r$. We find that for all $r \geq 2$, every $(r+1) \times (r+1)$ correlation matrix is $r$-decomposable, and we construct ${(2r+1) \times (2r+1)}$ correlation matrices that are not $r$-decomposable. One question this leaves open is whether every $4 \times 4$ correlation matrix is $2$-decomposable, which we make partial progress toward resolving. We apply our results to an entanglement detection scenario. Show Chinese abstract

Abstract: 相关矩阵（对角线为1的半正定矩阵）在量子信息理论中具有基础性的重要性。 在本工作中，我们引入并研究了$r$可分解的相关矩阵集合：那些可以表示为最多秩为$r$的相关矩阵的Schur乘积的矩阵。 我们发现，对于所有$r \geq 2$，每个$(r+1) \times (r+1)$相关矩阵都是$r$可分解的，并且我们构造了${(2r+1) \times (2r+1)}$相关矩阵，它们不是$r$可分解的。 一个未解决的问题是，每个$4 \times 4$相关矩阵是否都是$2$-可分解的，我们在这方面取得了部分进展。我们将结果应用于纠缠检测场景。