A graphical calculus for integration over random diagonal unitary matrices

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution random vectors containing independent uniform complex phases. Our method exploits order structure partially ordered set block permutations. A similar is developed consisting signs, based on combinatorics even partitions. employ our extend some results by Johnston and MacLean family local diagonal unitary invariant matrices. Furthermore, approach applies just as well real (orthogonal) case, where we introduce notion triplewise complete positivity study condition separability relevant bipartite Finally, analyze twirling linear maps between matrix algebras matrices, showcasing another application method.