On the zero entries in a unitary matrix

We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for unitary and orthogonal matrices are the same. They are both the set 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 orthogonal matrix of Schmidt rank 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].