ABSTRACT

We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for $n\times n$ unitary and orthogonal matrices are the same. They are both the set $\{0,1,\dots ,n^2-n-4,n^2-n-2,n^2-n\}$ for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation is a product operation. The latter is a fundamental problem in quantum information. We also construct an $n\times n$ orthogonal matrix of Schmidt rank $n^2-1$ with many zero entries, and it solves an open problem in Muller-Hermes and Nechita [Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 2018;557:174—187].

摘要

我们研究酉矩阵中零条目的数量。我们证明了零条目数的集合$n\times n$酉矩阵和正交矩阵是相同的。他们都是集合$\{0,1,\dots ,n^2-n-4,n^2-n-2,n^2-n\}$对于n > 4。我们用集合中的数字显式地构造正交矩阵的示例。我们应用我们的结果来构建一个必要条件，通过该条件，多方单一操作是一个产品操作。后者是量子信息中的一个基本问题。我们还构建了一个$n\times n$施密特秩正交矩阵$n^2-1$有许多零条目，它解决了 Muller-Hermes 和 Nechita 中的一个开放问题 [二部酉矩阵的运算符 Schmidt 秩。线性代数应用程序。2018；557:174—187]。