论文标题
在随机对角线矩阵上积分的图形微积分
A graphical calculus for integration over random diagonal unitary matrices
论文作者
论文摘要
我们提供了一个图形微积分，用于计算张量网络图的平均值，相对于包含独立统一复合阶段的随机向量的分布。我们的方法利用了部分有序的一组均匀块排列的顺序结构。基于部分有序的偶数分区集合的组合，为由独立统一符号组成的随机向量开发了类似的演算。我们采用我们的方法扩展了约翰斯顿和麦克林对当地对角线统一矩阵家庭的一些结果。此外，我们的图形方法同样适用于真实（正交）情况，在那里我们介绍了三方向完全阳性的概念，以研究相关的两部分矩阵可分离性的条件。最后，我们通过独立的对角线统一矩阵分析了基质代数之间线性图的旋转，展示了我们方法的另一种应用。
We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our graphical approach applies just as well to the real (orthogonal) case, where we introduce the notion of triplewise complete positivity to study the condition for separability of the relevant bipartite matrices. Finally, we analyze the twirling of linear maps between matrix algebras by independent diagonal unitary matrices, showcasing another application of our method.
