Local Unitary Equivalence of Orthogonal Arrays Based on Difference Matrices

In the investigation of quantum entanglement, local unitary (LU) equivalence serves as a fundamental tool for distinguishing and classifying entanglement structures. Orthogonal arrays are classical combinatorial designs with applications to quantum states and quantum error-correcting codes. Two orthogonal arrays are said to be LU equivalent if their associated quantum states are equivalent under LU operations. In this paper, we study the LU equivalence of three concrete families of orthogonal arrays constructed from mutually orthogonal Latin squares and difference matrices via Fourier matrices. We determine their strengths using projection maps and minimum support sizes. These constructions yield LU equivalent orthogonal arrays whose numbers of rows need not be the same, and in general, whose strengths may be different. Under a natural condition on the first local unitary, we further show that LU equivalence between two OAs obtained from difference matrices implies LU equivalence between their underlying difference matrices. These findings provide a new perspective for investigating the LU equivalence of orthogonal arrays and their underlying difference matrices, and reveal deep connections between combinatorial structures with different parameters.